r 


REESE  LIBRARY 

ov  THI-: 

UNIVERSITY  OF  CALIFORNIA 

,  i8(><?. 

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No. 


Accession  No.  J  5  0  6  5  "  •    Clam 


TEXT  BOOKS 

FOB 

ENGINEERS  AND  STUDENTS. 

BY 

DE    VOLSON    WOOD, 

Professor  of  Engineering  in  Stevens  Institute  of  Technology. 


A  TREATISE  ON  THE  RESISTANCE  OF  MATERIALS,  AND 
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SECOND  EDITION, 
REVISED  AND  ENLARGED. 


NEW  YORK  : 
JOHN  WILEY  &  SONS. 

LONDON: 
CHAPMAN  &  HALL,  LIMITED. 

1896. 


COPTRIGHT,   1896, 
BY 

DB  VOLSON  WOOD. 


PRINTED  BY  THE  BURR  PRINTING  HOUSE, 
FRANKFORT  AND  JACOB  STREETS,  NEW  YORK. 


REMARK. 

About  sixty-five  years  ago  M.  Poncelet  made  a  solution  of  the 
Fourneyron  turbine  which,  for  its  thoroughness  and  the  direct- 
ness of  its  analysis,  has  become  classical  (Comptes  Rendus,  1838). 
But  that  writer  neglected  the  frictional  (and  other)  resistances 
within  the  wheel,  and  assumed  that  the  buckets,  or  passages  in 
the  wheel,  were  constantly  full.  The  former  is  an  important 
element  in  the  theory,  and  its  consideration  makes  the  analysis 
but  little  more  complicated. 

Weisbach,  in  his  Hydraulic  Motors,  gives  a  solution  in  which 
frictional  resistances  are  involved,  and  the  sections  of  the  stream 
at  the  outlet  of  the  supply  chamber,  the  entrance  into  the  wheel, 
and  all  the  sections  of  the  buckets  are  determined  when  the 
wheel  runs  for  best  efficiency.  The  formulas,  however,  are  so 
complex  that  but  little  practical  knowledge  can  be  gained  from 
their  general  discussion.  I  have,  therefore,  assumed  that  the 
wheels  here  discussed  have  about  the  proportions  made  for  com- 
mercial purposes,  and  deduced  certain  numerical  results  which 
are  entered  in  tables  ;  and  a  simple  examination  of  these  furnishes 
certain  desirable  information. 

The  driving  power  here  considered  is  that  of  an  incompressible 
fluid,  which  in  practice  will  be  water.     The  steam  turbine  or 
those  driven  by  a  compressible  fluid  are  not  in  practice  con 
structed  like  water  turbines,  and  no  theory  for  such  is  here  at- 
tempted. 

The  first  part  of  this  work,  to  page  65,  develops  the  general 
theory  of  Turbines,  and  the  latter  part  treats  of  actual  wheels, 
their  forms,  construction,  capacity  and  efficiency. 

THE  AUTHOR. 


HYDRAULIC    MOTORS. 


A  general  solution  of  all  classes  of  turbines,  including  the  fric- 
tional  resistances,  is  here  attempted.  There  are  two  general 
classes :  one  in  which  the  water  enters  the  buckets  freely,  with 
a  velocity  due  to  the  head,  and  the  other  in  which  its  entrance  is 
more  or  less  resisted  by  the  pressure  in  the  buckets,  and  hence 
the  velocity  is  less  than  that  due  to  the  head.  The  former  are 
called  turbines  of  "free  deviation;"  the  latter,  " pressure"  tur- 
bines. 

There  will  first  be  given  a  general  solution  of  the  "  pressure 
turbine,"  and  the  other  turbines  will  be  considered  as  special 
cases  of  the  more  general  one. 

2.  Notation. 

Let  Q  be  the  volume  of  water  passing  through  the  wheel 
per  second, 

tf,  the  weight  of  unity  of  volume  of  the  water,  or  62^  pounds 
per  cubic  foot ;  then 

6Q=  W  will  be  the  weight  of  water  passing  through  the 
wheel  per  second, 

h}  be  the  head  in  the  supply  chamber  above  the  entrance 
to  the  buckets, 

A2,  the  head  in  the  tail  race  above  the  exit  from  the  bucket, 

Zj,  the  fall  in  passing  through  the  buckets, 

H^ki  +  Zi  —  h2y  the  effective  head, 

U,  the  useful  work  done  by  the  water  upon  the  wheel, 

R,  the  work  lost  by  frictional  resistances,  whirls,  etc., 

/ii,  the  coefficient  of  resistance  along  the  guides, 

/^2,  the  coefficient  of  resistance  along  the  buckets, 


HYDRAULIC   MOTORS. 


rb  the  radius  of  the  initial  rim, 
r2,  the  radius  of  the  terminal  rim, 
p,  the  radius  to  any  point  of  the  bucket, 
n=rl-^-r^  the  ratio  of  the  initial  to  that  of  the  terminal  radius, 
V,  the  velocity  of  the  water  issuing  from  supply  chamber, 
Vi9  the  initial  velocity  of  the  water  in  the  bucket  in  reference 

to  the  bucket, 

v,  the  velocity 
along  the  bucket 
at  any  point, 

vZ9  the  termi- 
nal velocity  in 
the  bucket, 

Fa,  the  velocity 
of  exit  in  refer- 
ence to  the  earth, 
™,  the  angular 
velocity  of  the 
wheel, 

«-,  terminal 
angle  between 
the  guide  and  in- 
itial rim  =  CA  B, 

yly  angle  between  the  initial  element  of  bucket  and  initial 
rim  =  EAD, 

y2  =  G-FI,  the  angle  between  the  terminal  rim  and  terminal 
element  of  the  bucket, 

0  =  HFI,  angle  between  the  terminal  rim  and  actual  direc- 
tion of  the  water  at  exit, 

p\9  the  pressure  of  water  at  entrance  of  the  bucket  per  unit 
area, 

p,  the  pressure  of  water  at  any  point  of  the  bucket, 

Pv  the  pressure  of  water  at  exit, 

pa,  the  pressure  of  an  atmosphere, 

a  =  eb,  the  arc  subtending  one  gate  opening,  Fig.  3, 


FIG.  1. 


RADIALLY  OUTFLOW  TURBINE. 


HYDRAULIC   MOTORS.  6 

a1?  the  arc  subtending  one  bucket  at  entrance.  (In  Fig.  3, 
a  and  a^  appear  to  be  the  same,  but  in  practice  they  are  usually 
different,  a  being  greater  than  ar) 

a^  =  gh,  the  arc  subtending  one  bucket  at  exit. 

7T,  normal  section  of  passage,  bf  being  its  projection,  it  be- 
ing assumed  that  the  passages  and  buckets  are  very  narrow  ; 

kl9  initial  normal  section  of  bucket,  Ijd  its  projection ; 

&2,  terminal  normal  section,  gi  being  its  projection ; 

Y,  the  depth  of  K,  yl  of  &„  and  yt  of  k. 
Then 


rim 


rj  =  velocity   of    initial 
^  —  velocity  of  terminal 


rim. 


FIG.  2. 


3.   General  Solution. 

Beginning  with  the  pres- 
sure on  the  top  of  the  supply 
chamber,  the  relation  between 
the  heads,  actual  and  virtual,  will  be  determined  to  the  point 
of  discharge  from  the  wheel. 

The  pressure  per  unit  on  the  upper  surface  of  the  supply 
chamber  will  be  that  of  the  atmosphere,  or 

Pa, 

and  the  corresponding  virtual  head  in  terms  of  a  column  of 
water  will  be 


The  head  in  the  supply  chamber  above  the  entrance  to  the 
wheel  will  be 


HYDRAULIC   MOTORS. 

therefore,  the  total  head  above  the  initial  element  of  the  bucket 
will  be 


This  head  produces  an  actual  pressure  p{  at  the  entrance 
to  the  bucket  and  the  velocity  V  of  exit  from  the  guides ; 
hence,  according  to  Bernoulli's  theorem,  the  heads  due  to  the 
pressure  pi  and  velocity  T7,  will  equal  the  former,  or 

pa       Pi         V2 
V+T  =  ~1T  +  1  T » (2) 


P 


(3) 


which  will  be  the  theoretical  pres- 
sure at  entrance  to  the  bucket  if 
friction  be  neglected.  Represent 
the  head  lost  by  friction  by 


which  must  also  be  overcome   by 
the  head  in  the  supply  chamber, 
so   that  we  have,  by  adding   it  to 
the  second  member  of  (2)  and  transforming, 


FIG.  3. 


(1  + 

The  triangle  of  velocities  ABC,  Fig.  1,  gives 

sin  1/1 


T;r  sn  yl 

V—    -  -  -  -  — 

Sin  (a  + 


--  — 

sin  (a  + 


sin  en 


(4) 


(5) 


The  relation  between  the  initial  and  terminal  velocities  in 


FOUKNEYRON   Tui'.HINE 


HYDRAULIC   MOTORS, 


the  bucket  involves  the  velocity  of  the  wheel  and  the  pressure 
in  the  bucket. 

Let  m  be  an  elementary  mass  at  a  distance  p  from  the  axis 
of  the  wheel,  then  will  the  centrifugal  force  be 


and  if  this  element  by  moving  a  distance  ds  in  the  tube  also 
moves  outward  a  distance  dp,  the  work  done  by  the  centrifugal 
force  will  be 


FIG.  4. 


If  the  tube  (or  bucket)  be 
inclined  downward,  the  work 
done  (or  energy  acquired)  by 
the  weight  in  falling  a  distance 
dz  will  be 

mgdz. 

These  two  works  will  be  ex- 
pended in  the  following  ways  : 

a.  Increasing  the  energy  of  the  water  in  the  tube  in  refer- 
ence to  the  tube  by  an  amount 

i  md  (v*). 

b.  In  doing  work  against  the  difference  of  pressures  on  the 
two  faces  of  the  element,  and  considering  the  back  pressure 
p  greater  than  the  forward  pressure  p,  the  work  will  be 

dp 


where  dp  —  3  is  equivalent  to  a  head  through  which  mg  would 
work. 

c.  In  overcoming  frictional  resistance.  The  law  of  fric- 
tional  resistances  is  not  well  known,  but  is  assumed  to  vary 
as  the  energy  of  the  mass  and  wetted  perimeter.  The  pe- 
rimeter is  here  discarded,  hence  the  work  will  be 


HYDRAULIC    MOTORS. 

>ua.-J  mv*ds. 
Hence  we  have 

mgdz  +  mo^pdp  =  %  md  (v*)  +  mg  »|-  +  \  ^mv*ds,     .    .    .    (7) 

But  the  last  term  cannot  be  integrated  unless  v  be  a 
known  function  of  s,  and  since  this  is  not  known,  we  make  v  = 
v2,  the  terminal  velocity.  The  coefficient  /<2  is  determined 
independently  of  the  length,  and  includes  the  value  ^9  when 
s  is  the  length  of  a  bucket. 

Integrating  between  initial  and  terminal  limits  gives 


The  fall  ^  is  so  small  in  practice  compared  with  the  next 
term  of  the  equation,  that  it  may,  and  will,  be  omitted,  giving 

{1  +  ^>  =  rf  +«»(,,>  _,fl_  ty  SLZ£,  .    .    .    (9) 

which  gives  v* 

At  exit  the  pressure  will  be 

(10) 


The  velocity  of  exit,  relative  to  the  earth,  will  be 

(11) 


The  work  done  upon  the  wheel  will  be  the  initial  (poten- 
tial) energy  of  the  water  less  the  energy  in  the  water  as  it 
quits  the  wheel,  still  further  diminished  by  the  energy  due  to 
frictional  losses  ;  or 


.....    (13) 


Three  tubes  having  small  orifices,  A,  B,  C,  on  the  same  arc,  ro- 
tate about  a  common  axis,  0,  one  discharges  to  the  left,  another 
radially,  and  the  third  to  the  right ;  they  receive  water  at  the 
ends  near  to  the  axis,  with  so  small  a  velocity  as  to  be  negligible  ; 
required  the  velocity  of  exit.  Equation  (9)  gives  for  all  three 
cases  v^  =  &?  r*  for  the  velocity  relative  to  the  tubes;  and 
equation  (11)  gives  for  1st,  F~2  —  2  GO  r^ 

"       «     2d,  7,  =  V2.a>r 

«      "     3d,  Fa  =  0. 


HYDKA.ULIC   MOTORS. 


111 

.s  9.2 


§ 


OD     S.     PH 


§  ° 

ry     cC 
cQ 


M 

-2 


o  .T! 

5    § 


41 


0.  .g    $ 


o  . 

-S    ^    "§ 

|a  s 

02    '5^     PH 


cur 


+ 

S!^ 

02 


•i  s 


— '    c? 
I         + 


'S      + 


&.    ! 
s  "   I 


•I 


1 


8 


HYDRAULIC   MOTORS. 


=  L  [— 
when 


cos 


.    .    .     (15) 


-Jf= 


-  1  + 


4-  Pi\ra 


2  cos  ex  sin 
sin  (a  +  ri)  sin2  (a  +  ^i) 


'       2 


\ 


(15a) 


For  maximum  efficiency  make  6?^  -r-  doo  =  0  in  (15)  and  solve 
for  GO,  calling  this  particular  value  GO',  then 


'I?'-*0?"'    •    •    •    (16) 
Ta 


which  value  substituted  in  equation  (15)  will  give  the  maxi- 
mum efficiency.     Then  equations  (5),  (6),  become 


-r-r 
V= 


sin 


sin  (a  -f  y 

sin  a 

sin  (a  +  ; 

Also  from  (9),  (10),  (4),  (17),  (18), 


GOT}, 


(17) 
(18) 


But 


sin2  or  — 


sin2(a  +  y\}         sin2o:  —  sinVi  +  2  sin  (a  +  y±)  cos  a  sin  ^t 
2  sin  (a  4-  ;KJ)  cos  a  sin  ^, 


=  1- 


2  cos  a  sin  y , 


sin  (a  +  ^,)  ' 
-which  substituted  above  will  give  equation  (14). 


Remark. 
Substituting  eq.  (16)  in  (15)  gives,  by  reduction, 

E™-=  v.  Ji-r^rW-  VJf'-^eosVJ 


(168) 


, 

7^*  tfA<3  terminal  angle  of  the  guide,  a,  the  initial  and  ter- 
minal angles  of  the  bucket  y1  and  yn  respectively,  the  ratio  of 
the  radii  r1  ?\  and  the  frictional  resistances  constant  y  then  for 
all  such  wheels  M  and  N  will  be  constant  and  the  efficiency  will 
be  constant  y  the  velocity  of  the  initial  rim,  QD'  /•„  the  velocity 
through  the  gate  V,  the  initial  velocity  in  the  bucket,  will  each 
and  all  vary  as  VH. 


HYDRAULIC  MOTORS. 


,,'-  2  rf-K   .  ,      ^  .    (19) 

sin  - 


The  normal  sections  of  the  buckets  will  be 


The  depths  of  those  sections  will  be 

77  Z.  If 

~\7~  1  2  /O1  \ 

J.  sin  « '      ^  ~  «i  sin  ^t '      ^2  "  <^2  sin  xa  ' 


DISCUSSION. 

4.   Three  simple  systems  are  recognized. 

rx  <  r2,  called  outward  flow. 

rx  >  r2,  called  inward  flow. 

r±  =  r2,  called  parallel  flow. 

The  first  and  second  may  be  combined  with  the  third,  mak- 
ing a  ra{#ed  system.  The  third,  in  theory,  is  really  an  inward 
or  outward  flow,  with  an  indefinitely  narrow  crown,  although 
the  analysis  applies  to  a  parallel  flow  wheel,  in  which  the 
width  is  indefinitely  small,  and  depth  small  compared  with 
the  total  head. 

5.    Value  of  /„  the  quitting  angle. 

Equations  (14)  and  (16a)  show  that  the  efficiency  is  increased 
as  cos  y2  is  increased,  or  as  y2  decreases,  and  is  greatest  for  y2=& 
Hence,  theoretically,  the  terminal  element  of  the  bucket  should 
be  tangent  to  the  quitting  rim  for  best  efficiency.  This,  how- 
ever, for  the  discharge  of  a  finite  quantity  of  water,  would  re- 
quire an  infinite  depth  of  bucket,  as  shown  by  the  third  of  equa- 
tions (21).  In  practice,  therefore,  this  angle  must  have  a  finite 


10  HYDRAULIC   MOTOES. 

value.  The  larger  the  diameter  of  the  terminal  rim  the  smaller 
may  be  this  angle  for  a  given  depth  of  wheel  and  given  quantity 
of  water  discharged.  Theoretical  considerations  then  would 
require,  for  best  efficiency,  a  very  large  diameter  for  the  quit- 
ting rim,  and  a  very  small  angle,  y2,  between  the  terminal  ele- 
ment of  the  bucket  and  the  rim  ;  but  commercial  considera- 
tions require  some  sacrifice  of  best  efficiency  to  cost,  so  that 
a  smaller  diameter  and  larger  angle  of  discharge  is  made.  If 
wheels  are  of  the  same  diameter  and  depth,  the  inward  flow 
wheel  requires  a  larger  quitting  angle  for  the  same  volume 
of  water  than  the  outward  flow,  since  the  discharge  rim  will  be 
smaller  in  the  former  than  in  the  latter  wheel,  and  the  velocity 
v2,  eq.  (19),  will  also  be  less.  In  practice  y2  is  from  10°  to  20°. 

6.  Relation  between  y2  and  GO'. 

Equation  (16)  when  put  under  the  form 


NT 


J  N'2  COS2  y 

* 


(22) 


shows  that  <*>'  increases  as  y2  decreases,  and  is  largest  for 
y2  =  0  ;  that  is,  in  a  wheel  in  which  all  the  elements  except  y2 
are  fixed,  the  velocity  of  the  wheel  for  best  effect  must  increase  as 
the  quitting  angle  of  the  bucket  decreases. 

If  the  terminal  element  be  radial,  then  y2  =  90°,  and  equa- 
tion (22)  appears  to  give  GO'  =  0  ;  but  the  discussion  really  fails 
for  this  case.  See  article  89. 

7.  Values  of  a  -f  y^ 

If  a  +  Kl  =  JL80°,  and  a  and  yl  both  finite,  then  will  If  and 
N  in  (15a)  both  be  infinite  ;  but  equation  (5)  gives 


that  is,  the  wheel  will  have  no  motion,  and  no  work  will  be 


FOURNEYRON  TRIPLE  WHEEL. 


*V~     OF   THF  '^iy 

UNIVEBSITY 


HYDRAULIC    MOTORS.  11 

done.  If  a  +  yl  =  180°,  then  the  terminal  element  of  the 
guide  and  the  initial  element  of  the  bucket  have  a  common 
tangent,  in  which  case  the  stream  can  flow  smoothly  from  the 
former  into  the  latter  only  when  the  wheel  is  at  rest.  (See 
Fig.  5.) 

If  a  4-  ;/j  exceed  180°,  GO'  would  be  negative,  and  it  would 
be  necessary  to  rotate  the  wheel  backwards  in  order  that  the 
water  should  flow  smoothly  from  the  guide  into  the  bucket. 

It  follows,  then,  that  a  -t-  y^  must  be  less  than  180°,  but  the 
best  relation  cannot  be  determined  by  analysis  ;  however, 
since  the  water  should  be  deflected  from  its  course  as  much  as 
possible  from  its  entering  to  its  leaving  the  wheel,  the  angle  a 
for  this  reason  should  be  as  small  as  practicable. 

8.   Values  of  a. 

If  a  —  0,  equation  (14)  will  reduce  to 


+  r2  cos  y*  V  ZgH  +  (rf  -  2n2  -  w?)  a       .   .     (24) 


which  is  independent  of  y}  ;  hence,  for  this  limiting  case,  the 
efficiency  will  be  independent  of  the  initial  angle  of  the  bucket. 
This  is  because  the  water  enters  the  wheel  tangentially  and 
therefore  has  no  radial  component  that  would  give  an  initial 
velocity  in  the  bucket  ;  and  equation  (18)  shows  that  the  ini- 
tial velocity  vl  would  be  zero,  while  (17)  shows  that  the  velocity 
of  the  initial  rim  must  equal  that  of  the  water  flowing  from  the 
guides,  or 

V—  GO'TI. 

For  the  limiting,  or  critical  case, 

a  —  0,  y2=  0,  Ui  —  0,  A/2  =  0, 

the  velocity  producing  maximum  efficiency  will  be,  from  equa- 
tion ^16), 

.......    (25) 


12 


HYDRAULIC   MOTORS. 


or  the  velocity  of  the  initial  rim,  if  the  wheel  be  frictionless, 
will  be  that  due  to  half  the  head  in  the  supply  chamber. 
If  r}  =  2n2,  then 

......    (26) 


or  the  velocity  of  the  terminal  rim  will  equal  that  due  to  the 
head.  Substituting  in  (19)  the  values  a  =  0,  y2  =  0,  /^  =  0, 
^2  —  0,  r}  =  Zr*,  and  it  will  reduce  to 


.......     (27) 

as  it  should. 

The  following  table  gives  the  values  of  quantities  for  the 
three  classes  of  wheels  : 

TABLE  I. 


72    = 


=0, 


=0. 


VELOCITI 

r  OF 

Velocity 
of  Exit 

VELOCITY 

IN  BUCKET. 

Velocity 

DIMENSIONS 
OF  WHEEL. 

Inner  Rim. 

Outer 
Rim. 

from 
Guide 
V. 

Initial 
t'j. 

Terminal 
ra. 

of  Exit. 
w. 

CL>>! 

GO'TZ 

f  i=V$r* 

r}=rz 

VffS 

yfi* 

V*ffH 

VffH~ 

VffH 
V&2 

0.00 
0.00 

^/Zgll 
VffZ 

0.00 
0.00 

1.000 
1.000 

Ti=lAr2 

GO'TZ 
0.714%/^ 

GO'TI 

V~gH 

Vffff 

0.00 

.nt^H 

0.00 

1.000 

In  the  first  case  the  inner  rim  is  the  initial  one,  in  the 
third  case  the  outer  rim  is  initial,  it  being  an  inward  flow 
wheel. 

Since,  in  this  case,  the  velocity  of  admission  to  the  wheel 
in  reference  to  the  earth  is  that  due  to  half  the  head  in  the 
supply  chamber,  and  the  velocity  of  exit  is  zero,  it  follows 
that  the  energy  due  to  the  velocity  is  all  imparted  to  the 


RADIALLY  INFLOW  TURBINE. 


HYDRAULIC   MOTORS. 

wheel  ;  and  the  energy  due  to  the  remaining  half  of  the  head 

is  imparted  to  the  wheel  by  pressure  in  the  wheel.     If  the 

velocity  of  entrance  to  the  wheel  be  that  due  to  the  head,  or 

F2  =  %gH  then  will  no  energy  be  imparted  to  the  wheel  on 

account  of  pressure  exerted  by  any  part  of  the  head  77,  but  if 

V-  <  2^/77,  then  will  some  of  the  work  be  done  by  this  press- 

ure, 10  being  zero.     For  the  cases  in  Table  I.,  the  energy  im- 

parted to  the  ivheel  ivill  be  due  one-half  to  velocity  and  one-half  to 

pressure  /  or  in  symbols, 


TF. 

+^WH=   Wff,    .     .     (28) 

or,  the  entire  potential  energy  of  the  water  will  be  expended 
in  work  upon  the  wheel. 

Whenever  F2  <  2(/77,  the  pressure  at  entrance  must  exceed 
the  external  pressure  at  exit,  and 

F- 
H  —  - 

_?2,   ........     (29) 

H 

then  will  be  the  part  of  the  head  producing  pressure  in  the 
wheel. 

In  practice,  a  cannot  be  zero  and  is  made  from  20°  to  30°. 
When  other  elements  of  the  wheel  are  fixed,  the  value  of  a 
may  be  determined  so  as  to  secure  a  certain  amount  of  initial 
pressure  in  the  wheel,  as  will  be  shown  hereafter. 

The  value  ?\  =  1.4r2  makes  the  width  of  the  crown  for 
internal  flow  about  the  same  as  for  i\  —  *Jfy\  for  outward  flow, 
being  approximately  0.3  of  the  external  radius. 

9.    Values  of  /./,  and  /<2. 

The  frictional  resistances  depend  not  only  upon  the  con- 
struction of  the  wheel  as  to  smoothness  of  the  surfaces,  sharp- 


14 


HYDRAULIC  MOTORS. 


ness  of  the  angles,  regularity  of  the  curved  parts,  but  also 
upon  the  manner  it  is  run ;  for  if  run  too  fast,  the  initial  ele- 
ments of  the  wheel  will  cut  across  the  stream  of  water,  pro- 
ducing eddies  and  preventing  the  buckets  from  being  filled, 
and  if  run  too  slow,  eddies  and  whirls  may  be  produced  and 
thus  the  effective  sections  be  reduced.  These  values  cannot 
be  definitely  assigned  beforehand,  but  Weisbach  gives  for 
good  conditions, 

^  =  ^  =  0.05  to  0.10..    .     .     .    . '   . .   (30) 

They  are  not  necessarily  equal,  and  //,  may  be  from  0.05  to 
0.075,  and  //2  from  0.06  to  0.10,  or  values  near  these. 

10.  Values  of  y^ 

It  has  already  been  shown  that  /,  must  be  less  than 
180°  —  (x.  If  y,  =  90°,  equation  (14)  shows  that  the  efficiency 
of  the  frictionless  wheel  will  be  independent  of  a.  The  effect 
of  different  values  for  yl  is  best  observed  from  numerical 
results  as  shown  in  the  following  table  : 

TABLE   II. 


Let     a  =  25°, 


=  12°, 


INITIAL 

ANGLE. 

yi« 

(i) 

TI  =  r^t. 

r,  =  1.4r2. 

«'ra. 

(2) 

y. 

(3) 

CO'TV 
(4) 

w'r2. 

(5) 

E. 

(6) 

"ffir 

60° 

1.322V<7# 

.812 

,934y7# 

.780^/^ff 

.911 

1.092-y/^ 

90° 

1.226     " 

.827 

.866     " 

.689     " 

.908 

.964     " 

120° 

1.078     " 

.838 

.762     " 

.576     " 

.898 

.806     " 

150° 

.518     " 

.744 

.366    " 

.271     " 

.752 

.379     " 

The  values  o?Va  in  columns  (2)  and  (5)  are  velocities  for  the 
terminal  rim,  which  in  column  (2)  are  for  the  exterior  rim,  but 


PARALLEL  FLOW  TURBINE,  GIRARD  TYPE. 


HYDRAULIC   MOTORS. 


15 


for  column  (5)  it  is  the  interior  rim,  while  column  (7)  is  for 
the  exterior  rim. 

Columns  (2)  and  (7)  show  that  the  velocity  of  the  outer 
rim  is  less,  for  maximum  effect,  for  the  inflow  than  for  the  out- 
flow, for  the  same  size  wheel. 

Column  (3)  shows  that  the  efficiency,  E,  decreases  as  the 
initial  angle  of  the  bucket,  ;/„  increases  up  to  120°.  This 
maximum  will  be  for  this  wheel  with  this  amount  of  friction. 

Column  (6)  shows  that  for  the  inflow  wheel  the  efficiency 
continually  decreases  as  ;/,  increases.  If  the  head  and  quan- 
tity of  water  discharged  be  constant,  the  work  would  be  pro- 
portional to  the  efficiency ;  for,  from  equation  (14), 

U=dQHE     .......     (31) 

The  effect  of  yl  on  the  velocities  is  shown  in  Table  III. 

TABLE   III. 


Let  a  =  25°, 


=  UZ  =  0.10, 


INI- 

r,w»: 

,,  =  >.4,, 

TIAL 
ANGLE. 

F 

»i 

»« 

K* 

ft,   x 

ft2  x 

F 

Vi 

& 

A'x 

ft,     X 

*2    X 

VgH 

Vffff 

VgH 

VgH 

VgH 

4/<7// 

4/^ 

VgH 

V^ 

VgH 

4/<7# 

VgH 

60° 

.820 

.396 

1.447 

1.2192.525 

.691 

.959 

.463 

.761 

1.043 

2.160 

1.314 

90° 

.955 

.403 

1.378 

1.0472.481 

.725 

1.063 

.449 

.676 

.940 

2.227 

1.479 

120° 

1.150 

.560 

1.153 

.8691.785 

.874 

1.217 

.593 

.605 

.821 

1.686 

1.653 

150° 

2.1001.775 

.621 

.476 

.563 

1.610 

2.060 

1.741 

.296 

.485 

.574 

3.378 

For  commercial  considerations  it  may  be  necessary  to  sac- 
rifice some  efficiency  to  save  on  first  cost,  and  to  avoid  making 
the  wheel  unwieldy. 

From  equation  (4)  it  appears  that  the  pressure  in  the  wheel 
at  entrance,  pl9  diminishes  as  the  velocity  of  admission,  V,  in- 


16  HYDRAULIC   MOTORS. 

creases,  and.  according  to  equation  (5),  V  depends  upon  y\ 
when  a  is  fixed.  Since  the  crowns  are  not  fitted  air  tight  nor 
water  tight  it  is  desirable  that  pl  should  exceed  the  pressure 
of  the  atmosphere  when  the  wheel  runs  in  free  air,  or  the  press- 
ure p2  +  pa  when  submerged,  to  prevent  air  or  water  from  flow- 
ing in  at  the  edge  of  the  crown.  It  will  be  shown  hereafter,  in 
discussing  the  pressures  in  the  wheel,  that  we  should  have 

-  tan  yl  >  tan  2^, (32) 

or,  180°    -  Yi>  2 of, 

or,  ;/!  <  180°  -  2«. 

If  a  =  30, 

then  yl  <  120°. 

To  be  on  the  safe  side,  the  angle  yl  may  be  20  or  30  degrees 
less  than  this  limit,  giving 

y1  =  180°  -  2«  -  25  (say) 
=  155     -  2«. 

Then  if  a  =  30°,  yl  =  95°.  Some  designers  make  this  angle 
90°,  others  more,  and  still  others  less  than  that  amount.  Weis- 
bach  suggests  that  it  be  less  so  that  the  bucket  will  be  shorter 
and  friction  less.  This  reasoning  appears  to  be  correct  for  the 
inflow  wheel,  for  the  size  and  conditions  shown  in  Table  II., 
but  not  for  the  outflow  wheel.  In  the  Tremont  turbines,  de- 
scribed in  the  Lowell  Hydraulic  Experiments,  this  angle  is  90°, 
the  angle  ^,  20°,  and  y2,  10°.  Fourneyron  made  yt  —  90°,  and 
n  from  30°  to  33°. 

In  Table  III.  it  appears  that  for  yl  =  150°,  V—  2.1  VgH, 
which  exceeds  VSgfB  $  that  is,  the  velocity  of  exit  from  the 
supply  chamber  exceeds  that  due. to  the  head,  hence  the  pressure 
at  entrance  into  the  wheel  must  be  less  than  that  of  the  atmo- 
sphere. For  zero  pressure  for  the  frictionless  wheel,  the  above 
condition  gives 

=  180°  -  2o 


PARALLEL  FLOW.     JONVAL  WITH  DRAFT  TUBE. 


HYDRAULIC    MOTORS.  17 

which  for  a  —  25°,  gives  j\  =  130 :,  and  for  f/t  ==  150°,  the  press- 
ure would  be  negative,  and  for  120 J  it  would  be  positive.  It 
appears  that  for  the  wheel  with  friction,  considered  in  the  table, 
that  this  pressure  is  also  positive  for  yl  —  120°,  and  negative 
for  150°. 

11.  Form  of  Bucket. 

The  form  of  the  bucket  does  not  enter  the  analysis,  and 
therefore  its  proper  form  cannot  be  determined  analytically. 
Only  the  initial  and  terminal  directions  enter  directly,  and  from 
these  and  the  volume  of  the  water  flowing  through  the  wheel, 
the  area  of  the  normal  sections  may  be  found  from  equations 
(20).  . 

But  well-known  physical  facts  determine  that  the  changes  of 
curvature  and  section  must  be  gradual,  and  the  general  form 
regular,  so  that  eddies  and  whirls  shall  not  be  formed.  For  the 
same  reason  the  wheel  must  be  run  with  the  correct  velocity 
to  secure  the  best  effect ;  for  otherwise  the  effective  angles  a 
and  YI  may  be  changed  to  values  which  cannot  be  determined 
beforehand,  in  which  case  the  wheel  cannot  be  correctly  ana- 


FIG.  6. 

lyzed.  In  practice  the  buckets  are  made  of  two  or  three  arcs 
of  circles  mutually  tangential  at  their  points  of  meeting.  Also, 
if  the  normal  sections,  K,  klt  &„,  of  the  buckets  as  constructed 
do  not  agree  with  those  given  by  computation,  the  stream  will, 
if  possible,  adjust  itself  to  true  conditions  by  the  formation  of 
2 


18 


HYDRAULIC   MOTORS. 


eddies.  If  the  terminal  sections  at  the  guides,  or  the  initial 
section  of  the  bucket,  be  too  small,  the  action  may  be  changed 
from  a  pressure  wheel  to  one  of  free  deviation.  So  long  as  the 
pressure  in  the  wheel  exceeds  the  external  pressure,  the  pre- 
ceding analysis  is  applicable  for  the  wheel  running  for  best 
effect,  observing  that  the  sections  K,  kjt.  &,,  are  not  those  of 
the  wheel,  but  those  which  are.  computed  from  the  velocities 
F,  vl9  vv 


12.    Value  of  0 ;  or  direction  of  the  quitting  water. 
From  Fig.  1  it  may  be  found  that 


and 


',  cos  B  =  v.2  cos  y-2  - 
r2  sin  6  =  v.2  sin  yz  ; 

.  cot  0  —  cot  y.>  —  — 


sin 


(33) 

(34) 

(35) 


These  formulas  are  for  the  velocity  giving  maximum  effi- 
ciency. If  the  speed  be  assumed,  GO  in  place  of  oo'  becoming 
known,  v-2  is  given  by  equation  (19).  It  is  apparent  for  such  a 
case  that  6  may  have  a  large  range  of  values  from  6  =  y2i  when 
the  wheel  is  at  rest,  to  6  exceeding  90°  for  high  velocities.  The 
following  table  gives  some  results  : 

TABLE   IV. 


a.  =  25°, 


=  12°, 


=//2  =  0.10. 


• 

TI  =  T*  4/t. 

r,«=1.4r, 

Yi 

0) 

•  6 

CO 

6 

60° 

.314  V^H 

72°  14' 

.160  VjH 

102°  43' 

90° 

.310  " 

66°  59' 

.143  " 

101°  17' 

120° 

.241  " 

60°  24' 

.126  " 

82°  52' 

150° 

.157  " 

55°  26 

.043  " 

74°  51' 

INFLOW  PARALLEL  CROWNS. 


HYDRAULIC   MOTORS.  19 

According  to  this  table  the  water  is  thrown  backward,  or 
in  the  direction  opposite  to  the  motion  of  the  wheel  for  the 
outward  flow  wheel,  and  for  the  inflow  it  is  thrown  forward  for 
Yi  less  than  90°,  and  backward  for  YI  greater  than  120°. 

In  the  Tremont  turbine  a  device  was  used  for  determining  the 
direction  of  the  water  leaving  the  wheel,  and  for  the  best  effi- 
ciency, 79J  per  cent,,  the  angle  6  was  about  120°.  Lowell 
Hydraulic  Experiments,  p.  33. 

The  angle  thus  observed  had  a  large  range  of  values  ranging 
from  50°  to  140°  for  efficiencies  only  two  or  three  per  cent,  less 
than  79^  per  cent. 

13.   Of  the  vahte  of  GO. 

So  far  as  analysis  indicates,  the  wheel  may  run  at  any  speed  ; 
but  in  order  that  the  stream  shall  flow  smoothly  from  the 
supply  chamber  into  the  bucket — thus  practically  maintaining 
the  angles  a  and  Y\. — the  relations  in  equations  (5)  and  (6)  must 
be  maintained,  or 

=  ^n^v          |  .I....     (36) 
sin  Y,. 

and  this  requires  that  the  velocity  V  shall  be  properly  regu- 
lated, which  can  be  done  by  regulating  the  head  hi  or  the  press- 
ure pi  or  both  hi  and  p{,  as  shown  by  equation  (4).  This 
however  is  not  practical.  In  practice,  the  speed  is  regulated, 
and  when  the  condition  for  maximum  efficiency  is  established, 
the  velocities  V  and  Vi  are  found  from  equations  (17)  and  (18). 
Since  ;/2,  in  practice,  is  small  we  have,  for  best  effect, 

v2  =  c»>2,  approximately,      ....     (37) 

and,  adopting  this  value,  a  more  simple  expression  may  be 
found  for  the  velocity  of  the  wheel.     For  equation  (19)  gives 

V?.  —  r*  GO' 

.  /^s 

=  .(Approx.)(38) 


A 


cos  asm  ylfr 


20  HYDRAULIC   MOTORS. 

If  Ml  =  ^  =  0.10,  r2  *  ?',  =  1.40,  «  =  25°,  yi  =  90°,  the  velo- 
city of  the  initial  rim  for  outward  flow  will  be 

VgH 

&}'?•,  =. —       y       n  Q9Q 

Vi  +  0.159  ~ 

The  velocity  due  to  the  head  would  be 


0.659   .....     .     (39) 


hence,  the  velocity  of  the  initial  rim  should  be  about 

°-928 
1.414 

of  the  velocity  due  to  the  head. 

For  an  inflow  wheel  in  which  r?  =  2r22,  and  the  other  dimen- 
sions, as  given  above,  this  becomes 


of  the  velocity  due  to  the  head. 

The  highest  efficiency  of  the  Tremont  turbine,  found  experi- 
mentally, was  0.79375,  and  the  corresponding  velocity,  0.62645 
of  the  velocity  due  to  the  head,  and  for  all  velocities  above  and 
below  this  value  the  efficiency  was  less.  Experiment  showed 
that  the  velocity  might  be  considerably  larger  or  smaller  than 
this  amount  without  diminishing  the  efficiency  very  much. 

In  the  Tremont  turbine  it  was  found  that  if  the  velocity  of 
the  initial  (or  interior)  rim  was  not  less  than  44  nor  more  than 
75  per  cent,  of  that  due  to  the  fall,  the  efficiency  was  75  per 
cent,  or  more.  Exp.,  p.  44. 

This  wheel  was  allowed  to  run  freely  without  any  brake 
except  its  own  friction,  and  the  velocity  of  the  initial  rim  was 
observed  to  be  1.335  V2gII,  half  of  which  is 

0.6675  VfyH,        ......    (41) 


THE 


THE  "  HERCULES." 
MIXED  FLOW,  INWARD  AND  DOWNWARD. 


HYDRAULIC   MOTORS. 


21 


"  which  is  not  far  from  the  velocity  giving  maximum  effect ; 
that  is  to  say,  when  the  gate  is  fully  raised  the  coefficient  of  effect  is 
a  maximum  when  the  wheel  is  moving  with  about  half  its  maximum 
velocity."  Exp.,  p.  37. 

M.  Poncelet  computed  the  theoretical  useful  effect  of  a 
certain  turbine  of  which  M.  Morin  had  determined  the  value  by 
experiment.  The  following  are  the  results  (Comptes  Rendus, 
1838,  Juillet) : 

TABLE  V. 


Velocity  of  initial 

rim  or 

7'ito' 

Number  of  turns  of 
the  wheel  per 

Ratio  of  useful  to 
theoretical  effect. 

Means  of  values  by 
experiment. 

^2gH~ 

minute. 

0.0 

0.00 

0.000 

0.2 

33.80 

0.<>64 

0.4 

47.87 

0.773 

0.700 

0.6 

58.61 

0.807 

0.705 

0.7 

62.81 

0.810 

0.700 

0.8 

67.67 

0.806 

0.675 

1.0 

75.76 

0.786 

0.610 

1.2 

82.88 

0.753 

0.490 

1.4 

89.52 

0.712 

0.360 

1.6 

95.70 

0.664 

0.280 

1.8 

101.51 

0.612 

0.203 

2.0 

107.00 

0.546 

0.050 

3.72 

145.00 

0.000 

Poncelet  states  that  he  took  no  account  of  passive  resist- 
ances, and  hence  his  results  should  be  larger  than  those  of 
experiment  as  they  are ;  but  here  both  theory  and  experiment 
give  the  maximum  efficiency  for  a  velocity  of  about  0.6  that 
due  to  the  head,  and  the  efficiency  is  but  little  less  for  velocities 
perceptibly  greater  and  less  than  that  for  the  best  effect.  For 
velocities  considerably  greater  and  less,  theoretical  results  are 
much  larger  than  those  found  by  experiment,  for  reasons 
already  given,  chief  of  which  is  the  fact  that  eddies  are  induced, 
and  the  effective  angles  of  the  mechanism  changed  to  unknown 
values. 


22  HYDRAULIC   MOTORS. 

14.  Pressure  in  the  wheel. 

Dropping  the  subscript  2  from  v,  r,  p,  in  equation  (9),  the 
resulting  value  of  p  will  give  the  pressure  per  unit  at  any  point 
of  the  bucket  providing  that  /^2  be  considered  constant.  Chang- 
ing r  to  p,  equation  (9)  thus  gives 

— i  jf 

-•    (42) 

To  solve  this  requires  a  knowledge  of  the  transverse  sec- 
tions of  the  stream,  for  the  velocity  v  will  be  inversely  as  the 
cross  section. 

From  equations  (20)  and  (6) 

ft,      ft,        sin. 


From  (4)  and  (5), 

.  1  +  u,         sin2 


!  a  ^  . 

2^r        sin2  (a  + 

These  reduce  equation  (42)  to 


•         (44) 


-  i  [( 


1  +  ^sinV,  +  [fc       4-         - 


The  back  or  concave  side  of  the  bucket  will  be  subjected 
to  a  pressure  which  may  be  considered  in  two  parts  :  one  due 
to  the  deflection  of  the  stream  passing  through  it,  the  other  to 
a  pressure  which  is  the  same  as  that  against  the  crown,  and  is 
uniform  throughout  the  cross  section  of  the  bucket,  due  to  the 
pressure  of  a  part  (or  all)  of  the  head  in  the  supply  chamber. 
It  is  the  latter  pressure  which  is  given  by  the  value  of  p  in 
equation  (45).  The  construction  of  the  wheel  being  known, 
the  pressure  p  may  be  found  at  any  point  of  the  wheel  for  any 
assumed  practical  velocity  ;  although,  for  reasons  previously 


SEGMENTAL  FEED.     RADIALLY  INWARD  FLOW.     TANGENTIAL  WHEEL. 


HYDKAULIC    MOTORS.  23 

given,  it  will  be  of  practical  value  only  when  running  near  the 
velocity  for  maximum  efficiency.     "  There  are  two  cases  : 

1.  That  in  which  the  discharge  is  into  free  air  ; 

2.  That  in  which  the  wheel  is  submerged.  * 

In  the  first  case  if  the  pressure  is  uniform,  the  case  is 
called  that  of  'free  deviation'  in  which  the  entire  pressure 
upon  the  forward  side  of  the  bucket  is  due  to  the  deviation  of 
the  water  from  a  right  line,  and  will  be  considered  further  on. 

If  equation  (45)  shows  a  continually  decreasing  pressure 
from  the  initial  element  to  that  of  exit,  or  if  the  minimum 
pressure  exceeds  pa,  the  preceding  analysis  is  applicable. 
But  if  it  shows  a  point  of  minimum  pressure  less  than  pa,  it 
will  be  in  a  condition  of  unstable  equilibrium,  in  which  the 
slightest  inequality  would  cause  air  to  rush  in  and  restore  the 
pressure  to  that  of  the  atmosphere  ;  so  that  the  pressure  in 
the  wheel  and  the  flow  would  be  changed.  The  point  of  mini- 
mum pressure  may  be  found  by  plotting  results  found  from 
equation  (45),  substituting  values*  for  p  taken  from  measure- 
ments of  the  wheel,  and  ~k  from  computation.  From  the 
entrance  of  the  wheel  up  to  the  point  of  minimum  pressure 
the  preceding  analysis  applies  ;  and  the  remainder  of  the  wheel 
must  be  analyzed  for  '  free  deviation  '  and  the  two  results  added. 

In  the  second  case  the  equations  will  apply,  since  air  can- 
not enter,  provided  that  p  does  not  become  negative,  to  realize 
which  requires  a  tensile  stress  of  the  water.  This  is  impos- 
sible and  eddies  would  be  formed  ;  and  the  effect  of  these  on 
the  velocity  and  pressure  cannot  be  computed.  Such  a  case 
cannot  be  analyzed." 

15.  To  find  the  pressure  at  the  entrance  to  the  bucket  when 
running  at  best  effect.  In  (45)  let  p  —  r,,  k  =  ki  and  p  —  p\. 
To  simplify  still  more,  let  the  wheel  be  frictionless,  or  ^  = 
H2  =  0,  and  find  from  equation  (38) 

«    ,9       sin  (a  +  i/ 
2 


cos  of  sm 


,  . 
(46) 


24  HYDRAULIC   MOTORS. 

also  hi  =  H  +  A-2,  and  (45)  becomes 

Pl  =  611+  6h*  +  pa  -  d* 


- 
2  cos  a  sin  («  +  xO 

If  the  wheel  is  not  submerged  h.2  =  0,  and  let  the  pressure 
PI  equal  that  of  the  atmosphere,  or  pa,  then 

-  sinri 


2  cos  (v  sin  (a  + 


If  the  wheel  be  submerged,  let  p{  —  <Sh2  +  pa,  and  the  equa- 
tion reduces  to  that  of  the  preceding. 
Equation  (48)  gives 

tan  2<-v  =  —  tan  yl9 
or,  2<*  =  180°~x1;      ....     .     (49) 

for  which  value  the  pressure  at  the  entrance  to  the  wheel  will 
equal  that  just  outside. 

If,  2^>  180°  --  ylt     .     .     .     .     .     .     (50) 

the  pressure  within  will  be  less  than  that  without  ;  but  if 

2«  <  180°  --  xi,      ..-     .-    .'.    (51) 

the  pressure  within  will  exceed  that  without  —  a  condition 
which  is  considered  desirable.  If  frictional  resistances  be 
considered  the  value  of  r^ri  from  equation  (38)  will  be  less 
than  that  given  by  (46),  and  hence  the  last  term  of  equation 
(47)  will  be  less  unless  a  be  greater  than  the  value  given  by 
equation  (51)  ;  hence  with  frictional  resistances  the  terminal 
angle  of  the  guide  blade  may  exceed  somewhat  90°  -  ^yl  ; 
therefore,  if  the  value  of  a  be  found  for  a  frictionless  wheel  it 
will  be  a  safe  value  when  there  is  friction.  If  y  =  90°  and 
a  =  90°  -  £xi  =  45°>  tnen  (47)  gives 

8h,+pa,       .    .     (52) 


SCOTTISH  on  WHITELAW. 


BARKER  MILL. 
WHEELS  WITHOUT  GUIDES. 


•«|P  i 

V^       or  TH* 

UNIVERSITY 


HYDRAULIC    MOTORS.  25 

as  it  should.     If  yl  =  90°  and  a  —  30°,  then 


or,  p,  >  6k,  +pa  +  0.33tf//.     .     .     ....     .    (53) 


The  angle  a  should  not  be  so  small  or  y\  so  small  as  to  pro- 
duce excessive  pressure  at  the  entrance  to  the  wheel. 

Example.  —  Find  the  pressure  per  square  inch  at  the 
entrance  to  the  wheel  when  the  head  is  10  feet,  the  terminal 
angle  of  the  guide  is  30°,  the  initial  angle  of  the  bucket 
yl  =  90°  ;  the  wheel  being  one  foot  under  the  water  in  the  tail 
race. 

16.  Number  of  buckets. 

The  analysis  given  above  is  true  for  a  wheel  with  a  single 
bucket,  provided  the  supply  is  constantly  open  to  the  bucket 
and  closed  by  the  remainder  of  the  wheel.  But  for  practical 
considerations  the  wheel  should  be  full  of  buckets,  although 
the  number  cannot  be  determined  by  analysis.  Successful 
wheels  have  been  made  in  which  the  distance  between  the 
buckets  was  as  small  as  0.75  of  an  inch,  and  others  as  much  as 
2.75  inches.  Lowell  Hyd.  Exp.,  p.  47.  Turbines  at  the  Centen- 
nial Exposition  had  buckets  from  4V  inches  to  9  inches  from 
centre  to  centre. 

17.  Ratio  of  radii. 

Theory  does  not  limit  the  dimensions  of  the  wheel.  In 
practice, 

for  outward  flow,  r2  -f-  ^  is  from  1.25  to  1.50  {          ,~. 
for  inward  flow,  r2  -f-  rl  is  from  0.66  to  0.80  ) 

It  appears  from  Table  II.  that  the  inflow  wheel  has  a 
higher  efficiency  than  the  outward  flow  wheel  (columns  6  and 
3),  and  these  wheels  have  about  the  same  outside  and  inside 
diameters.  The  inflow  wheel  also  runs  somewhat  slower  for 


26  HYDRAULIC    MOTORS. 

best  effect.  The  centrifugal  force  in  the  outward  flow  wheel 
tends  to  force  the  water  outward  faster  than  it  would  other- 
wise flow;  while  in  the  inward  flow  wheel  it  has  the  contrary 
effect,  acting  as  it  does  in  opposition  to  the  velocity  in  the 
buckets. 

It  also  appears  that  the  efficiency  of  the  outward  flow  wheel 
increases  slightly  as  the  width  of  the  crown  is  less,  and  the 
velocity  for  maximum  efficiency  is  slower  ;  while  for  the  inflow 
wheel  the  efficiency  slightly  increases  for  increased  width  of 
crown  and  the  velocity  of  the  outer  rim  at  the  same  time  also 
increases. 

Let  r,  =  nrt,  Yl  =  90°,  y,  =  20°,  /i,  =  ^  =  0,  a  =  80°  ; 

then  for  n  =  0,  0.5,  0.8,  1.4, 

we  have  GJ'T,  =  0,  0.761^//,  0.972^  #",  1. 

coV1  =  1.3910//,  1.52%/#,  1.215*7 //,  0. 
E  =  0.6594,      0.8050,      0.9070,     0.9784. 

18.  Efficiency,  E. 

The  method  of  determining  the  theoretical  value  of  E  has 
already  been  given ;  but  to  determine  the  actual  value,  resort 
must  be  had  to  experiments.  These  have  been  made  in  large 
numbers  and  the  results  published.  By  assuming  the  mini- 
mum values  of  the  several  losses,  a  maximum  limit  to  the  effi- 
ciency may  be  fixed.  Thus,  if  the  actual  velocity  be  0.97  of 
the  theoretical,  the  energy  lost  will  be  (1-0.972)  or  6  per  cent. 
Friction  along  the  buckets  and  bends  .  .  .  5  "  " 

Energy  lost  by  impact,  say 2     "      " 

Energy  lost  in  the  escaping  water       ....     3     "      " 

Total     ....   16     "      " 
Leaving 84     "      " 

available  for  work,  This  discards  the  friction  of  the  mechan- 
ism and  frictional  losses  along  the  guides,  and  if  2  per  cent. 
be  allowed  for  the  latter,  there  will  be  left  82  per  cent.  It 


WHEEI 
JET  DIRECT-  ACTING. 


WHECL      IS 

SHOWN    MOUNTE1D    ON 

TEMPORARY  TRLSTLES 


JET  REACTION 
JET  WHEELS. 


HYDBAULIC   MOTORS.  27 

seems  hardly  possible  for  the  effective  efficiency  to  exceed  82 
per  cent.,  and  all  claims  of  90  or  more  per  cent,  for  these 
motors  should  be  at  once  discarded  as  being  too  improbable 
for  serious  consideration.  A  turbine  yielding  from  75  to  80 
per  cent,  is  extremely  good.  The  celebrated  Tremont  turbine 
gave  79 f  per  cent.  Lowell  Exp.,  p.  33.  Experiments  with 
higher  efficiencies  have  been  reported.  A  Jonval  turbine 
(parallel  flow)  was  reported  as  yielding  0.75  to  0.90,  but  Morin 
suggested  corrections  reducing  it  to  0.63  to  0.71.  (Weisbach, 
Mech.  of  jEng.,  vol.  ii.,  p.  501.)  Weisbach  gives  the  results  of 
many  experiments,  in  which  the  efficiency  ranged  from  50  to 
84  per  cent.  See  pages  470,  500-507.  See  also  Jour.  Frank. 
Inst.,  1843,  for  efficiencies  from  64  to  75  per  cent.  Numerous 
experiments  give  E  =0.60  to  0.65.  The  efficiency,  considering 
only  the  energy  imparted  to  the  wheel,  will  exceed  by  several 
per  cent,  the  efficiency  of  the  wheel,  for  the  latter  will  include 
the  friction  of  the  support,  and  leakage  at  the  joint  between 
the  sluice  and  wheel,  which  are  not  included  in  the  former  ;  also 
as  a  plant  the  resistances  and  losses  in  the  supply  chamber  are 
to  be  still  further  deducted. 

19.  The  Crowns. — The  crowns  may  be  plane  annular  discs, 
or  conical,  or  curved.   If  the  partitions  forming  the  buckets  be 
so  thin  that  they  may  be  discarded,  the  law  of  radial  flow  will 
be  determined  by  the  form  of  the  crowns.     If  the  crowns  be 
plane,  the  radial  flow  (or  radial  component)  will  diminish  as 
the  distance  from  the  axis  increases — the  buckets  being  full — 
for  the  annular  space  will  be  greater. 

20.  Designing. 

The  dimensions  of  a  wheel  must  be  determined  for  a  definite 
velocity.  Thus  far  it  has  been  assumed  that  the  angles  ar,  y}> 
etc.,  are  given,  and  the  normal  sections  of  the  stream  thus 
deduced.  We  will  now  assume  that  all  the  dimensions  of 
the  buckets  are  known,  and  the  angle  a  and  the  section  JTare 


28  HYDRAULIC    MOTORS. 

to  be  determined.     The  velocities  vl  and  v9  must  now  be  found 
independently  of  a.     From  Fig.  1  we  have 

F2  =  v*  -\-  G^V!*  --  2vlGorl  cos  y\  (55) 

which  combined  with  equations   (4r),  (9),  (10),  vfa  —  vjc»  as  in 
(20),  and  If  =  /^  —  A2,  will  give 

A\  (1  +  Mi)  kji\(&)i\  cos  Y\ 


+  /*,)  V  +  J*M          '  L  (i  +  k)X'2  +  Ma  J 

•      .      -   '  (56) 
-      •        (56a) 


where 

^  _  (1  -h  /^i)  ^i^'i  eos  7i, 


Equations  (11),  (12),  (13),  (55),  (56),  and  (57),  after  making 


a  = 
I 
give 


=  1+A  +  /V'  I 

=  ^a  cos  7,  4-  //^j  cos  y.      } 


-  aA)  ooV^HB  +  O'oi]  •  (58) 


Remark. 

Since  jnl  is  a  comparatively  small  fraction  and  kl  exceeds  &2 
certain  terms  may  be  omitted  giving  for  v^  the  approximate  but 
very  nearly  exact  value, 


.    ,         cog 


I2 


and  corresponding  reductions  in  A,  B,  J. 


HYDRAULIC    MOTORS.  29 

Let 

D  =  gll  (1  4  aB), 

^  =  a  U"  +  *  S)  -  I  A  +  *  rf  +  i  ^S 
G  —  I  —  aA. 
Then 


•     (59) 
For  a  maximum  rZ^E7  -f-  r/cj  =  (9, 


This  value  of  co  is  the  one  to  be  used  in  the  other  equations. 
The  form  of  equation  (60)  is  the  same  as  that  of  equation  (16). 
Substituting  in  (59),  observing  that  aB  =  1,  and  hence  I)  =  0, 
we  find 


j  ^-3 


(61) 


(600) 


To  find  the  terminable  angle  a  of  the  guide  blade  that  will 
enable  the  stream  to  flow  smoothly,  subject  to  the  preceding 
conditions.  Fig.  1  gives 

Trcos  a  —  GOJ\  —  t\  cos  y^ 
which,  combined  with  equation  (55),  gives 

cos  a  =  *"''  "  *»  COS  ^  .  (62) 

Vv?  -f-  c^X2  —  2^,0?^  cos  yl 

Eliminating  vl  by  means  of  equations  (57)  and  (56)  gives 
cos  a  in  terms  of  the  six  constants  rjy  r^  yl  y»  Jcl  and  #2,  which 
are  fixed  and  known  from  the  dimensions  of  the  wheel,  and  of 
the  velocity  GO  of  the  wheel.  Since  the  wheel  may  run  at  dif- 


30 


HYDRAULIC   MOTORS. 


ferent  velocities  the  angle  «  must  vary,  and  this  will  be  done 
in  practice  by  the  piling  of  the  water  in  the  passages.  Each 
turbine,  however,  should  be  designed  to  run  at  the  speed  giv- 
ing maximum  efficiency,  and  its  angles  and  dimensions  should 
satisfy  equations  (60)  and  (62). 
From  equation  (9), 

+  6Jh    .    (63) 

in  which,  if  v2  and  v\  be  substituted  from  above,  pl  becomes 
known. 

Similarly,  V  from  equation  (55)  becomes  known,  and  finally, 
from  (60), 


aorl  —  vl  cos 
cos  a  =  — J 


(64) 


2L  Path  of  the  Water. — Let  aA  be  the  position  of  the  bucket 

when  the  water  enters  at  b. 
The  bucket  being  drawn  in 
position  to  a  scale,  divide  it 
into  any  number  of  parts — 
equal  or  unequal — aa,,  a^, 
etc.,  and  find  the  time  re- 
quired for  it  to  go  from  a  to 
Ox.  The  distance  being  small, 
assume  that  the  velocity  is 
uniform  from  a  to  alt  and 
equal  to  -y,,  which  will  be 
given  by  equation  (20), 

O 


FIG.  7. 


or  better, 


(64«) 


THE 

COLLI  NS 
WHEEL 


THE 

KNIGHT 
WHEEL 


HYDRAULIC   MOTORS.  31 

Then  will  the  time  t  be 

t  =  ^ (65) 

V 

During  this  time  the  rim  has  gone  from  a  to  b  a  distance 

ab=?\cot (66) 

If  the  bucket  bB  be  drawn  through  b,  and  the  arc  at  bi 
through  al5  their  intersection  h  will  be  the  position  of  the 
particle  at  the  end  of  the  time  t.  In  a  similar  manner,  the 
successive  points  Z>2,  ba,  etc.,  may  be  found,  through  which  a 
continuous  curve  may  be  drawn  representing  the  path  of  the 
stream. 

The  line  tangent  to  the  termination  of  the  bucket,  will 
indicate  the  direction  of  the  water  at  entrance  of  the  wheel, 
and  if  the  water  drives  the  wheel,  the  path  should  be  entirely 
outside  this  line  and  convex  toward  it. 

22.  Design  a  guide  blade,  outflow  turbine. 

Assume  the  effective  fall, H  —  ft. 

Assume  the  required  horse-power,    .     .     .  HP  = 

Assume  the  exit  angle,     .     .     .     ...     .     y-2  = 

Assume  the  entrance  angle  of  bucket,   .     .     y\  =  90°. 

Fix  the  exit  angle  of  the  guide,  Eqs.  (32),  (51),    a  =  30°.  ? 

Assume  efficiency,  .     .     .     .     .     .     .    .»     .     E=  0.65,  or  0.70; 

and  after  the  wheel  is  fully  designed,  re- 
compute this  value  and  if  necessary 
correct  the  dimensions. 

Required   quantity   of   water   per    second 

without  loss, .   •  • ; .      Q  —  U —  6H. 

Required  quantity,  .     .     ..     .     .     .       Q  -4-  E '  = 

Assume  ^  =  0.10,  /*2  =  0.075. 

Velocity  of  the  initial  rim,  Eq.  (40),  approx.,  ovr\  = 
(The  corrected,  final  value  will  be  found 
by  Eq.  (16)  or  (60). 


32  HYDKAULIC   MOTORS. 

Let  r.2  =  1.3rb  then  velocity  of  outer  rim,  oor2  ~ 
The  velocity  into  the  bucket,  Eq.  (18),  .  »  1%  » 
Initial  section  of  buckets,  Eq.  (20),  .  .  .  ki  = 
The  inner  circumference  will  be  2^.  Let 

the  walls  of  the  buckets  be  TV  of  the 

circumference,  then  the  effective  open- 

ings will  be  f|  of  the  circumference,  or 

i-^nr^     Assume    a   depth,    y,  between 

the  crowns.     Try  y  —  Jrx.     Then  will 

the   initial   cross    section   of    all    the 

buckets  be  \\nrf  ;  hence,  xt77"^2  —  &i  5  •"•  f\  = 
If  the  radius  is  not  what  is  desired,  it  may 

be  changed  to  some  other  value    and 

the  depth  y  computed.  Then,  ...  r2  = 
The  cross  section  at  outer  rim  will  be,  if 

the  crowns  are  planes,  ^7rr2  y  sin  y2  —  k2  = 
The  number  of  buckets  will  be  assumed  .  = 

Having  determined  these  elements,  the  final  velocity  v2  in 
reference  to  the  bucket  may  be  computed  by  equation  (67), 
G0i\  from  (62),  V  from  (55),  and  a  from  (64). 

If  the  turbine  revolves  in  air,  at  least  half  the  depth  of  the 
wheel  is  to  be  deducted  from  the  head  H. 

If  the  circular  opening  between  the  wheel  and  gate  be  ^ 
of  an  inch,  or  y|^-  of  a  foot,  and  the  coefficient  of  discharge  be 
0.7,  the  discharge  will  be 


0.7  x  2w,  x  i  =  q,          .    .    .    (67) 


PI  being  determined  from  equation  (63).     The  loss  of  work 
will  be 


62.2g  x  //    or     623g(H-$y)  .....     (68) 
The  work  lost  by  friction,  if  the  radius  of  the  axle  be  rs,  the 


TWIN  TURBINES  ON  A  HORIZONTAL  Axis, 


OF  THE 

UNIVERSITY 


HYDRAULIC  MOTORS.  33 

weight  of  the  loaded  wheel,  FF3,  and  coefficient  of  friction  //„, 

will  be  nearly 

o 

g  fo  TF3  r3cj  per  sec (69) 

The  work  done  by  the  water  must  be  the  effective  U  plus 
the  work  due  to  the  losses.  The  work  done  by  the  water  pass- 
ing through  the  wheel  must  be  U  plus  that  given  by  equation 
(69).  Call  this  C7j.  Compute  the  work  done  by  the  water,  and 
let  it  be  U% ;  then  will  the  required  depth  be 


The  efficiency  may  now  be  recomputed. 

The  following  tables,  YI.  and  YIL,  though  for  wheels  of 
special  dimensions,  give  some  general  results  as  shown  in  the 
following  "Conclusions."  The  sections  1',  and  &,  are  assumed 
to  be  those  of  the  wheel,  and  further  it  is  assumed  that  the 
wheel  passages  (or  channels)  are  filled  with  flowing  water,  and 
hence  without  eddies. 


SZ  1-1 


If. 

V     » 


00= 


«       "       2,       8 

d         d          d         ,-1* 


oooo 


S         fe         I          £ 


>  ^  > 

O     05       05 


» 


l'Esrl 


S  S    So    S  co    S  » 
ddoo    i-5  d    i-^d 


PIS 


H,<? 

V  V 


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e 


i-Jr-c    i-id    I-HO    od 


lffM£l*U 


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o   es     CJD   o     i-   «     i-    >S 

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000 

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ns  ss  ^s  &§ 

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d  d  do'  d  d  do 


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m%  *&  m  m 

w  v5  v  >  v  > 


1-1  O   00   00   00 


1111 

ddoo 


HYDRAULIC    MOTORS.  35 

Conclusions. — From  an  examination  of  Tables  II.,  III.,  YL, 
Vli.,  the  following  conclusions  are  drawn : 

1.  The  maximum  theoretical  efficiency  of  the  inflow  wheel  is 
perceptibly  larger  than  that  of  the  outflow,  the  width  of  crowns 
and  the  initial  and  terminal  angles  of  the  buckets  being  the  same. 
One  reason  for  this  is  due  to  the  flow  through  the  wheel  being 
opposed  by  the  centrifugal  action,  but  more  particularly  to  the 
smaller  velocity  of  discharge  from  the  inflow  wheel. 

2.  Columns  (10)  in  Tables  YI.  and   VII.  show  that  for  the 
wheels   here  considered   the  loss  of  energy  due  to  the  quitting 
velocity  is  from  2.2,  5.1  per  cent,  from  the  outflow,  and  from  0.9 
to  1  per  cent  for  the  inflow. 

3.  The  same   tables   show   that  in  column  (2)    the   efficiency 
is  almost  constant  for  the  varying  conditions  here  considered,  while 
for  the  outflow  there  is  considerable  variation. 

4.  One  of  the  most  interesting  and  profitable  studies  to   the 
theorist  and  practitioner  is  the  effect  upon  the  efficiency  due  to 
properly  proportioning  the  terminal  angle,  ex,  of  the  guide  blade. 
It  will  be  observed  that  all  the  efficiencies  in  Tables  YI.  and  VII. 
exceed  the  corresponding  ones  in  Table   II.   except  the  first  in 
column  (3)  of  Table  II.     In  Table  II.  the  terminal  angle,  a,  is 
constantly  25°,  while  in  Tables  VI.  and  VII.  it  is  less  than  that 
value,  and  in  the  highest  efficiencies  very  much  less. 

5.  It   appears   from  these  tables   that   the  terminal  angle,  <*, 
has  frequently  been  made  too  large  for  best  efficiency. 

6.  That  the  terminal  angle,  a,  of  the  guide  should  be  compara- 
tively small  for  best  effect ;  for  the  inflow  less  than  10°,  and  that 
theoretically,  when  the  angle  is  about  7°,  the  efficiency  is  some  10 
per  cent,  greater  than  when  it  is  25°  in  the  wheels  here  considered. 

7.  Tables  II.  and  VI.  indicate  that  the  initial  angle  of  the 
bucket  should  exceed  90°  for  best  effect  for  outflow  wheels. 

8.  Tables  II.  and  VII.  show  that  the  initial  angle  should  be 
less  than  90°  for  best  effect  on  inflow  wheels,  but  that  from  60° 
to  120°  the  efficiency  varies  scarcely  1  per  cent. 


36 


HYDRAULIC    MOTORS. 


9.  The  most  marked  effect  in  properly  proportioning  the  ter- 
minal angle,  «,  of  the  guide  is  shown  when  the  initial  angle  of 
the  bucket  is  1 50°.     In  this  case  the  efficiency  for  the  outflow 
when  a  is  25°  is  0.7-14,  Table  II.,  but  when  a  is  13£°,  as  in  Table 
YL,  it  becomes  0.921.     For  the  inflow,  in  the  former  case,  it  is 
0.752,  but  when  the  angle   is  3°,  as  in  Table  YIL,  it  becomes 
0.918. 

10.  Since  the  wheels  here  considered  have  the  same  width  of 
crowns  and  the  same  terminal  angle  of  the  bucket,  the  depths  of 
the  wheels  will  be  proportional  to  £2  for  discharging  equal  vol- 
umes of  water.    Tables  III.,  YL,  YII.  show  that  the  section  &a  in- 
creases as  the  initial  angle  of  the  buckets  increases,  and  that  it  must 
be  greater  for  the  inflow  than  for  the  outflow  ;  hence  the  depth 
of  the  wheel  must  be  greater  for  the  inflow  for  delivering  the 
same  volume  of  water. 

11.  But  the  same  volume  of  water  delivered  by  the  inflow  does 
more  work  than  that  of  the  outflow ;  the  depths  should  be  as  &,, 
divided  by   the  efficiency.     Thus  in  Tables  YL  and  YIL,  for 
y  =  90°,  and  for  the  same  heads,  II,  the  relative  depths  should 
be  for  equal  works  (0.759  ^-  0.828)  -j-  (150  -j-  0.920)  =  1.67. 

12.  In  the  outflow  wheel,  column  (9),  Table  YL,  shows  that  for 
the  outflow  for  best  effect  the  direction  of  the  quitting  water  in 
reference  to  the  earth  should  be  nearly  radial  (from  76°  to  97°), 
but  for  the  inflow  wheel  the  water  is  thrown  forward  in  quitting 
(column  [9]  Table  YIL).     This  alone  shows  that  the  velocity  of 
the  rim  should  somewhat  exceed  the  relative  final  velocity  back- 
ward in  the  bucket,  as  shown  in  columns  (1)  and  (5). 

13.  In  these  tables  I  have  given  all  the  velocities  in  terms  of 
V  2  g  h,  and  the  coefficients  of  this  expression  will  be  the  part 

of  the  head  which  would  produce  that  velocity  if  the  water  issued 
freely.  In  Tables  YL  and  YIL  there  is  only  one  case,  column 
(5)  of  the  former  table,  where  the  coefficient  exceeds  unity,  and 
the  excess  is  so  small  it  may  be  discarded ;  and  it  may  be  said 
that  in  a  properly  proportioned  turbine  with  the  conditions  here 


CASCADE  WHEEL. 


HYDRAULIC    MOTORS. 


37 


given,  none  of  the  velocities  will  equal  that  due  to  the  head  in 
the  supply  chamber  when  running  at  best  effect. 

14.  The  inflow  turbine  presents  the  best  conditions  for  con- 
struction for  producing  a  given  effect,  the  only  apparent  disad- 
vantage being  an  increased  first  cost  due  to  an  increased  depth, 
or  an  increased  diameter  for  producing  a  given  amount  of  work. 
The  larger  efficiency  should,  however,  more  than  neutralize  the 
increased  first  cost. 

15.  Column  7  and  equation  (29)  show  that  the  pressure  at  the 
initial  rim  decreases  as  the  initial  angle  yl  increases. 

16.  Tables  VI.  and  VII.  are  for  parallel  crowns.     Examples  of 
buckets  of  variable  depths  will  be  given  later,  and  are  illustrated 
in  Figs.  19  and  24. 

SPECIAL    WHEELS. 

23.  Fourneyron  Turbine. — All  wheels  having  guide  blades 
are  of  the  Fourneyron  type,  although  the  wheels  made  by  him 
were  outward  flow.  The  preceding 
analysis  is  a  general  solution  of  this 
turbine. 


24.  Francis  and  Thomson's  vortex 
viheels  are  inward  flow  wheels  with 
guide  blades.  The  preceding  analysis 
is  also  applicable  to  these  wheels. 


FIG.  8. 


25.  The  Jonval  Turbine  is  a  parallel 
flow  wheel  with  guide  blades,  to  which 

the  preceding  analysis  is  applicable  by  making  rt  —  r2. 

(For   the   details   of    these   and    many   other   forms,   see 
Hydraulic  Motors  by  Weisbach.) 

26.  Rankine  Wheel.— This   is   a  wheel  of  the  Fourneyron 
type,  but  Rankine  having  made  certain  modifications  in  its 


38  HYDRAULIC   MOTORS. 

assumed  construction  it  is  indicated  by  his  own  name. 
(Fig.  8.) 

It  is  an  outflow  wheel  and  the  crowns  are  so  made  that 
the  radial  velocity  of  the  water  in  passing  through  the  wheel 
will  be  uniform.  If  x  be  the  abscissa  from  the  axis  of  the 
wheel  to  any  point  of  the  crown,  and  y  the  distance  between 
the  crowns  at  that  point,  vr  the  radial  component  of  the 
velocity,  then 

y  •  27raj  -  vr  =  Q, 

or,  yx  =  a  constant,  ....     (71) 

which  is  the  equation  to  an  hyperbola  referred  to  its  asymp- 
totes. This  determines  the  form  of  the  crowns.  If  the  wheel 
were  inward  flow,  the  depth  would  be  greatest  at  the  inner 
rim. 

In  this  wheel  the  initial  element  of  the  bucket  is  radial,  or 
yl  =  90°  ;  and  Rankine  assumed  that  the  velocity  for  best  effect 
must  be  such  that  the  water  will  quit  the  wheel  radially,  or 
0  —  90°.  These  conditions  given,  from  Fig.  1  and  equations 
(5),  (6),  (34),  for  &  frictionless  wheel, 

G^T!  =  V  cos  a.  .     .     .....     (72) 

<jor2  =  v-2  cos  y2  .......     (73) 

Vi  =  w  —  Fsin  a  =  GOT^  tan  a  —  ojr2  tan  y2  =  v2  sin  y2',    (74) 

/.   tan  a  =  -  tan  ;/2,    ......     (75) 

which  determines  the  proper  angle  of  the  guide  blade  when 
the  value  of  y2  has  been  assigned.  If  y2  =  15°,  rz  =  IJri,  then 
a  =  18J°  ;  and  if  yz  =  20,  r2  =  1.3,  then  a  =  24°  nearly.  But 
to  be  certain  that  the  internal  pressure  exceeds  the  external, 
a  should  not  exceed  these  values.  Equations  (19),  (73),  and  (75) 
give 


which  establishes  the  velocity  of  the  initial  rim  of  the  wheel. 


Y      6- 


PLAN  OF  A  PROPOSED  FOURNEYRON  WHEEL. 


HYDRAULIC   MOTORS.  39 

The  work  in  the  frictionless  wheel  will  be  the  theoretical 
work  the  water  could  do,  less  the  energy  in  the  water  quitting 

the  wheel,  or 

W 

<y 

[Eq.  (74)]  =WH-%—  rfcf  tea*  a.    ...    (77) 


The  efficiency  will  be 


(79) 


27.  The  following  is  to  show  that  Kankine's  assumption  of 
velocity  for  best  efficiency  is  not  quite  correct.  Substituting 
Vl  =  90°,  ywi  =  0,  yw2  =  0,  n  —  nr.2,  in  equation  (16)  gives 


VH  r1  -  ^  -  v^1  -  *•)  2  -  °°sv«  (i  -  2rc2n 

--T^w\_-     V(l-rc2)2-cos2r2(T^2)       J 


^ 

and  these  in  equation  (19)  will  give 
m  — 


2^  1 
J 


-  n      -  cos  y,      - 
-  G,9r2  [1  -  n2  +  V  (1-  n2)  2  -  cos2  r2  (1  -  2/i2)J.     .     (81) 
This  does  not  give 


COS      2  — 


as  in  equation  (73),  except  when  n2  —  £  (or  2r2!  =  r^)  ;  and  hence 
the  direction  of  the  water  at  exit  will  not  be  radial,  as  Kankine 
assumed,  except  for  this  case  ;  and  hence  the  analysis  in  article 
26  is  not  applicable  to  the  inflow,  but  for  such  wheels  as  have 
the  proportion  2/1,2  =  /y2,  it  is  sufficiently  exact  for  the  friction- 


HYDRAULIC    MOTORS. 


less  outflow  wheel ;  and,  as  seen  above,  the  hypothesis  greatly 
simplifies  the  analysis. 

The  condition  for  best  efficiency  of  the  frictionless  wheel 
requires  that  the  velocity  of  leaving  the  wheel  should  be  a 
minimum  ;  and  this  may  be  realized,  in  some  cases,  when  its 
direction  is  oblique  to  the  radius. 

Thus,  let  AC  be  radial  when  AB  is  the  velocity  relative  to 
the  bucket,  and  BC  the  velocity  of  the  rim  ;  then  it  may  be,  in 
some  cases,  that  when  AD  is  the  relative  velocity  of  exit,  AEt 
the  velocity  of  exit  relative  to  the  earth,  will  be  less  than  A  Ct 
as  shown  in  Fig.  9. 


B 


FTG.  9. 


FIG.  10. 


28.  The  path  of  the  water  is  easily  constructed  for  this  wheel. 
Since  the  radial  velocity  is  uniform,  the  time  of  flowing  through 
the  wheel  will  be 


(82) 


during  which  time  the  initial  rim  AS,  Fig.  10,  will  have  travelled 

aB  =  oor^t  feet.       .     .     ....     (83) 

Divide  r>2  —  r{  into  equal  parts  by  concentric  arcs,  and  the 
space  aB  into  the  same  number  of  equal  parts,  and  through 
the  points  of  division  a,  &,  c,  d,  trace  the  buckets  ;  then  will  aD, 
drawn  through  the  proper  intersections  of  the  arcs  and  buckets, 
be  the  required  path. 


VERTICAL  SECTION  OF  A  PROPOSED  DOUBLE  FOUBNEYRON  WHEEL  WITH 
TRIPLE  CHAMBERS  FOR  GREAT  FALLS.     TRIPLE  DOUBLE  WHEEL. 


HYDRAULIC   MOTORS.  41 

9.  Analyze  a  Kankine  turbine,  having  given  :  H  =  12  feet, 

X2  --  15°,  >'i  =  2  feet,  ?'22  =  2rv    Depth  of  outer  rim,  6  inches. 

Find  Eadius  of  outer  rim,      ......    r2  = 

Angular  velocity,       .......    &>  = 

Velocity  of  initial  rim,       ....     ^co  = 

Velocity  of  outer  rim,    .....     r9a>  = 

Angle  of  guide  plates,    ......    a  = 

Velocity  from  the  supply  chamber,     .   V  = 
Initial  velocity  in  bucket,       .     .     .     .  Vi  = 

Terminal  velocity  in  bucket,  .     .     .     .  v2  = 

Velocity  of  quitting  water,      .     .     .     .    w  = 

Depth  of  inner  rim,  .  -  .     .    ,     .     .     .  y,  = 
The  horse-power,      ......     HP  = 

The  efficiency,  .........    E  =• 

If  the  partitions  for  the  buckets  occupy  ^  of  the  wheel,  and 

the  losses  due  to  frictional  resistances  in  the  wheel  and  friction 

of  the  wheel  be  20  per  cent.,  what  will  be 

The  horse-power,      ......      HP  = 

The  efficiency,       ........    E  = 

Find  the  pressure  at  the  inner  rim,    .     .     .    pi  = 

Find  the  path  of  the  water. 

29.  Velocity  of  a  particle  along  a  tube  rotating  about  an  axis 
perpendicular  to  its  plane. 

This  problem  has  already  been  solved  in  establishing  the 
general  equations  of  turbines,  and  the  following  is  given  to 
present  it  from  another  point  of  view. 

If  the  particle  at  A,  whose  mass  is  m,  be  confined  while  the 
tube  rotates  about  0,  Fig.  11,  with  the  angular  velocity  GO,  the 
centrifugal  force  would  be 

.......     (84) 


If  6  be  the  angle  between  the  radius  vector  prolonged  and 
the  normal  upon  the  tangent,  the  component  in  the  direction 
of  the  tangent  to  the  tube  will  be 


42  HYDRAULIC   MOTORS. 

moo2p  sin  6, 

and  when  the  particle  is  free  to  move,  this  component  will  be 
effective  for  producing  motion,  and  if  the  pressures  at  the 
opposite  ends  of  the  element  are  not  equal,  but  differ  by  an 
amount  dp,  we  have  the  equation 


cPs 

-T    =  moo2p  sin  0  —  dp. 


(85) 


D 


PIG.  12. 


But  cfesin  6  =  dp, 

which  combined  with  the  preceding  equation  gives 

dsd*s  1     dp     , 

,,2     =  co2pdp ; — -a  dp.     .    . 

dt2  m  sin  6  r 


(86) 


dp_ 
sin 
d 


But  -^-7,  =  dSj  and  if  6  be  the  weight  of  unity  of  volume, 

.  u 


then  m  =  -ds,  and  the  last  term  becomes  ^  dp.     The  integral 

J/ 

will  be 


.     (87) 


HYDRAULIC   MOTORS.  13 

If  the  friction  be  ^2v22,  the  equation  becomes 

.      (88) 

which  is  the  same  as  equation  (9). 
If  p2  —  Ph  and  fa  =  0>  then 

v}  =  V  +  *92  (ra*  -  rf)  ......     (89) 

This  gives  the  velocity  relative  to  the  tube  whether  it 
revolves  to  the  right  or  left,  and  whatever  be  its  curvature.  If 
it  revolves  to  the  left,  the  resultant  velocity  will  be  AD,  Fig. 
12  ;  if  to  the  right,  it  will  be  A  C.  If  y2  be  measured  from  the 
arc  backward  of  the  motion,  or  y%  =  BAF  for  rotation  to  the 
left,  and  y2  =  EAB  for  motion  to  the  right  ;  then 


AD2  =  w2-  =  v}  +  Go-rf  —  2v2  .  oor2  cos  yz.      .     .     (90) 
-  =  tv2  =  v2   +  co2r22  4-  2v2  .  oor2  cos  y2.      .     .     (91) 


In  the  latter  case  the  quitting  velocity  will  exceed  the  ter- 
minal velocity  in  the  tube,  and  therefore  increased  velocity  will 
have  been  imparted  to  the  water  —  a  condition  requiring  that 
energy  be  imparted  to  the  wheel  from  an  external  source.  Im 
the  former  case  the  wheel  is  a  motor,  in  the  latter  it  is  a  re- 
ceiver or  transmitter  of  power  ;  in  the  former  the  water  drives: 
the  wheel,  in  the  latter  the  wheel  drives  the  water  and  virtually 
becomes  a  centrifugal  pump. 

If  the  water  issues  tangentially  to  the  path  described  by  the 
orifice,  then  y2  —  0,  and 

F2  =  v2  T  Gar*  .     ......     (92) 

the  upper  sign  belonging  to  the  motor,  and  the  lower  to  the 
pump. 

Exercise.  —  If  ^  =  1  ft.  r2  =  5  ft.  /v2  =  0.1,  vl  =  5  ft.  per  sec- 
ond, and  the  bucket  rotates  about  a  vertical  axis  30  times  per 


44  HYDRAULIC   MOTORS. 

minute,  and  discharges  the  water  directly  backward,  making 
Yi  =  0,  required  the  terminal  velocity  along  the  tube  and  the 
velocity  of  discharge  relatively  to  the  earth. 

30.  Wheel  of  Free  Deviation. — In  this  wheel  the  water  in  the 
buckets  has  a  free  surface,  or,  in  other  words,  is  subjected  only 
to  the  pressure  of  the  atmosphere.  For  this  case 

Pz  =  Pi  =  pa  5  *i  =  H,  and  h-2  =  0, 
and  equation  (4)  gives 

(1  +  ,1,)  F2  -  20/7, (93) 

which   will  be   the   velocity   of    discharge   from   the   supply 
chamber  into  the  wheel ;  it  is  the  velocity  due  to  the  head  in 
the  supply  chamber  when  frictional  resistance  is  included. 
The  triangle  ABC,  Fig.  1,  gives 

V?  =  F2  +  a?  rf  -  2  Ffi>n  cos  a,         ...      (94) 
which  substituted  in  equation  (88)  gives 

(1  +  /<2)  v22  =  F2  +  GJ>  r}  -  2  F.  cor,  cos  a,       .     (95) 

and  this  in  equation  (90)  gives  F2,  and  equation  (12)  will  give 
the  required  work.  An  exact  general  solution  involves  a  solu- 
tion of  the  general  equation  of  the  fourth  degree.  See  article  89. 
'The  following  is  an  approximate  solution. 

If  ya  be  small,  and  the  wheel  be  run  for  best  effect,  that  is, 
so  as  to  make  the  velocity  F2  very  small,  and  considering  y2  =  0, 
equation  (92)  makes 

V2  =  cor2  nearly. 

Using  this  value  as  if  it  were  the  exact  one,  also  neglecting 
friction,  (95)  gives 

2  V.  vr}  cos  a  =   F2  = 
or  2^!  cos  «  = 


HYDRAULIC   MOTORS. 


which  gives  the  proper  velocity  of  the  initial  rim  ;  and  for  the 
terminal  rim 


cos 


Number  of  revolutions  per  minute 


(97) 


(98) 


To  find  the  velocity  at  any  point  of  the  bucket  relative  to  the 
bucket,  drop  the  subscript  2  from  equation  (95)  giving 


(1  +  //2)  v2  = 


-  2 


cos  a.   .     (99) 


FIG.  14. 


FIG.  13. 
From  Fig.  1  or  equations  (4)  and  (5)  find 


sin  yl  = 


sm  a. 


In  the  Motionless  wheel,  the  work  done  will  be 

and  the  efficiency  will  be 

U 


(100) 

(101) 
(102) 


31.  To  find  the  form  of  the  free  surface,  let  the  bucket  be  very 
narrow,  so  that  a  normal  to  one  of  the  curves  will  be  approx- 
imately normal  to  the  other.  Divide  one  side  of  the  bucket 


46  HYDRAULIC   MOTORS. 

into  any  convenient  number  of  parts,  as  «c,  ce,  etc.,  and  erect 
normals  to  the  arc,  as  ah,  cd,  etc.  Lay  off  these  arcs  on  a  right 
line.  Compute  the  velocity  at  any  point,  as  d9  Fig.  13,  by  for- 
mula (99).  Let  x  be  the  required  depth  at  d,  then  because  the 
velocity  into  the  section  equals  q,  the  volume  passing  through 
one  of  the  buckets  per  second,  we  have 

x.dc.v  =  q; 


dc.v 


(103) 


and  similarly  for  all  other  sections.  If  only  relative  heights 
are  to  be  found,  the  quantity  q  need  not  be  found,  for  if  y  be 
the  height  at  b,  Fig.  14,  then 

y.ba.Vl  =  q\ 
ba.vl 

' 


and  by  assuming  any  arbitrary  value  for  y  the  relative  value  of 
x  becomes  known.  Similarly,  the  relative  heights  at  all  other 
sections  may  be  found. 

32.  To  find  the  path  of  the  fluid  in  reference  to  the  earth,  pro- 
ceed as  in  Article  21  of  the  discussion  of  the  general  case. 

33.  Exercise.  —  Design  a  30  horse-power   inflow  turbine   of 
free  deviation,  given  an  effective  head  of  16  feet. 

Assume  the  depth  of  gate  opening  to  be  4  inches  (J  foot), 
and  after  the  computation  has  been  completed  if  it  does  not 
give  30  horse-power  the  depth  may  be  changed  by  proportion. 
Let  the  radius  of  the  outer  or  initial  rim  be  1  ft.  ;  of  the  inner 
rim,  f  of  a  foot  ;  terminal  angle  of  the  bucket,  y2  =  15°  ;  termi- 
nal angle  of  the  guide,  a  =  30°,  ^  =  0.10  —  /*2. 

Then,  velocity  of  exit  from  supply  chamber, 
Eq.  (93),       ....     ......      V= 

Velocity  of  outer  rim,  Eq.  (96),      .     .     .     .  <&ri  = 

Velocity  of  inner  rim,  Eq.  (97),      «.. 


HYDRAULIC    MOTORS. 


47 


Number  of  turns  per  minute, = 

Initial  angle  of  bucket,  Eq.  (100),  .  .  .  yl  = 
Initial  velocity  in  bucket,  Eq.  (94),  ...  vl  = 
Terminal  velocity  in  bucket,  Eq.  (95),  .  .  v2  = 

Velocity  of  exit,  Eq.  (90), w  = 

Direction  of  outflow,  Eq.  (35),  .  ....  6  = 
Coefficient  of  discharge  0.60,  volume  of 

water, Q  = 

Weight  of  water  (S  =  62.4),        .     .     . '  .    ,  dQ  = 

Work  per  second,  Eq.  (101), Z7  = 

Horse-power, HP  = 

Efficiency,        .     . E  — 

If  90  per  cent,  of   U  is  effective  work,  and  if  this  does  not 
give  30  horse-power,  then  the  depth  of  the  wheel  should  be 

30 

4  inches. 


Find  the  profile  of  the  stream  in  the  buckets. 

34.  The  following  is  taken  from  the  report  of  the  Commis- 
sioners of  the  Centennial  Exposition,  1876,  on  Turbines,  Group 
XX.  The  tests  were  for  two  minutes  each.  The  revolutions 
and  horse-powers  here  given  are  those  corresponding  to  the 
best  efficiencies  : 


Diameter 
of  wheel. 
Inches. 

Head  in  sup- 
ply chamber. 
Feet. 

Revolutions 
per  minute. 

Horse- 
power. 

Efficiency, 
per  cent. 

No.  of 
Buckets. 

Kind  of  wheel. 

30 

31 

255 

95 

85.0 

10 

Inflow. 

24 

31 

302 

67 

77.0 

14 

Parallel. 

24 

30* 

310 

64 

74.5 

13 

27 

30 

291 

76.8 

80.3 

16 

30 

30 

257 

74 

75.5 

18 

25 

31 

288 

46 

82.0 

12 

Parallel. 

30 

29.2 

258 

80.5 

78.7 

13 

In  and  down 

25 

30 

279 

62.5 

83.7 

15 

In  and  down. 

27 

30.4 

246 

53.2 

73.6 

14 

Parallel. 

36 

29.6. 

197 

66.2 

83.8 

26 

Parallel. 

HYDRAULIC   MOTORS. 


These  tests  were  by  no  means  exhaustive.  It  is  not  known 
that  they  were  run  for  best  effect.  The  distance  from  centre 
to  centre  of  buckets  varied  from  4.3  inches  to  9.5,  and  at  these 
extreme  values  the  efficiencies  were  about  the  same.  The 
number  of  gate  openings  was  less  than  the  number  of  buckets. 

TUKBINES  WITHOUT  GUIDES. 

35.  Barker's  Mill.  —  As  ordinarily  constructed,  this  motor 
has  two  hollow  arms  connected  with  a  central  supply  chamber, 

with  orifices  near  their 
outer  ends  and  on  oppo- 
site sides  of  the  arms. 
There  are  no  guide 
plates  The  supply 
chamber  rotates  with 
the  arms.  The  arms 
may  be  cylindrical,  con- 
ical, or  other  convenient 
shape. 

Since  the  water  issues 
perpendicularly    to    the 

arms  y-2  =  0  ;  and  since  the  initial  elements  of  the  arms  are 
radial,  yi  =  90°,  and  as  the  water  must  flow  radially  into  the 
arms,  a  —  90°.  The  inner  radius  is  necessarily  small  and  may 
be  considered  zero.  Hence,  making 

y,  =  Q,ri  =  90°,  a  =  90°,  n  =  0, 


FIG.  15. 


equation  (14)  gives 
U 

=w&= 

Equation  (19)  gives 


.     .-    (105) 


HYDRAULIC   MOTORS.  49 

hence,  the  efficiency  reduces  to,  for  the  frictionless  wheel, 


E  =  2  (107) 

V2  4-  oor.z 

This  has  no  algebraic  maximum,  but  approaches  unity  as 
the  velocity  increases  indefinitely.  Practically  it  has  been 
found  that  the  best  effect  is  produced  when  the  velocity  of  the 
orifices  is  about  that  due  to  the  head,  or 


.     .     .     .     •     •     (108) 
for  which  value  the  efficiency  will  be,  if  yw2  =  0.10 

=0.70.     .    .    .    .     (109) 


If  &2  be  the  area  of  the  effective  section  of  the  orifice,  then 

.     .    .    ,    .<•.'.."  (110) 


The  pressure  on  the  back  side  the  arms  opposite  the  orifices 

will  be 


.    ,    ....  (Ill) 

£/ 

Of  this  pressure  there  will  be  required 

P2  =  M.Gor2=^.oor2,     .....    (112) 

to  impart  to  the  water  the  rotary  velocity  oor2  which  it  has 
when  it  reaches  the  orifice.  The  effective  pressure  will  be 
PI  —  P2,  and  the  work  done  per  second  will  be  this  pressure 
into  the  distance  it  traverses  per  second,  or 

U=oor,[Pl  -P2], 

which  reduces  to  the  value  found  from  equations  (105)  and  (106). 

36.   Exercise.  —  Let  the  supply  chamber  be  square,  and  from 
two  of  its  opposite  sides  let  pyramidal   arms   project.     Let 


50  HYDRAULIC   MOTORS. 

H—  10  feet,  orifices  each  2  square  inches,  vertical  section  of 
arms  through  the  orifice  each  4  square  inches,  section  of  the 
arms  where  they  join  the  supply  chamber  each  8  square  inches, 
horizontal  section  of  the  supply  chamber  36  square  inches, 
r2  =  36  inches,  velocity  of  the  orifice  oor2  =  V^glf,  coefficient 
of  discharge  0.64,  and  //2  =  0.10. 

Required : 

Velocity  of  discharge  relative  to  the  orifice,  v-2  = 

Velocity  of  discharge  relative  to  the  earth,  Vt  — 

Velocity  at  entrance  to  the  arms,     ...  Vi  = 
Velocity  in  the  supply  chamber,      .     .     . 

The  volume  of  water  discharged,     ...  Q  = 

The  weight  of  water  discharged,     .     .     .  6Q  = 

The  work  per  second, 17= 

The  horse-power,       ...     .     .     ...    HP  = 

The  efficiency, E '  = 

The  pressure  on  arm  opposite  orifice  at  A 

per  square  inch, pl  = 

The  pressure  at  base  of  the  arms  at  C,     .  p  = 
The  equation  to  the  path  of  the  fluid. 

37. .  Scottish  and  Whitelaw  Turbines. — These  wheels  have  no 
guide  plates,  and  differ  from  Barker's  mill  chiefly  in  having 
curved  arms.  The  analysis  is  precisely  the  same  as  for  the 
Barker  mill.  The  only  practical  difference  consists  in  pro- 
viding a  curved  path  for  the  water,  instead  of  compelling  the 
water  to  seek  its  path,  forming  eddies,  etc. 

38.  Jet  Propeller. — We  first  show  how  this  problem  may  be 
solved  by  the  preceding  equations,  and  afterwards  make  an 
independent  solution.  Let  a  narrow  vessel,  Fig.  16,  be  carried 
by  an  arm  E  about  a  shaft  BA.  Let  water,  by  any  suitable 
device,  be  dropped  into  the  vessel,  the  horizontal  velocity  of 
the  water  being  the  same  as  that  of  the  vessel.  At  F,  the 
lower  end  of  this  chamber,  let  there  be  an  orifice  from  which 
water  may  issue  horizontally.  The  water  may  then  be  con- 


HYDRAULIC   MOTORS. 


51 


sidered  as  entering  the  vessel  or  bucket  without  velocity,  and 
passing  downward  finally  curve  towards,  and  issue  from,  the 
orifice.  It  thus  becomes  a  parallel  flow  wheel  without  guides, 
and  we  have,  for  tlie  frictionless  wheel, 


-  r2,     Yl  =  90°,     7/2  =5  0, 

#J  =  Pi   =  Pa, 


=  //2  -  0, 


in  equation  (8) ;  hence,  the  velocity  of  exit  relative  to  the  ori- 
fice will  be 


v}  =  20*2 


(113) 


FIG.  17. 


FIG.  16. 

where  %  equals  the  head  in  the  supply  chamber.  Under  these 
conditions  the  velocity  of  discharge  will  be  independent  of  the 
velocity  of  rotation,  if  the  rotation  be  uniform. 

Equation  (11)  gives  for  the  velocity  of  discharge  relative  to 
the  earth 

T7"  /  1  -i    A\ 

K2=  v2  —  cor2 (114) 

Equations  (19)  and  (14)  give 


=  — -  V2 
9 


(115) 


52  HYDRAULIC  MOTORS. 

<f  S~) 

This  equation  may  be  factored  thus,  —  —   is  the  mass  of 

iy 

liquid  flowing  out  per  second;  represent  by  MI  Mv-2  is  the 
momentum  of  the  outflowing  liquid  per  second.  Mv.2r2  is  the 
moment  of  the  momentum,  and,  finally, 

Mv2  •  T2co  is  the  moment  of  the  momentum  into  the  angular 
velocity,  and  equals  the  work  done. 

Let  v  =  Gor2  =  the  velocity  of  the  vessel  ;  then  from  (114) 
and  (115) 

W  =  V2  —  V,     ....... 


U,  =  Mv2v;    ........     (117) 

which   equations   are    true  whether  the   motion  be   circular, 
linear,  or  in  any  other  path. 

In  practice,  the  velocity  of  the  jet  is  produced  by  the  press- 
ure exerted  by  a  pump,  in  which  case  z2  in  equation  (113) 
would  be  replaced  by  a  virtual  head,  Fig.  17,  equivalent  to 
z-2  ;  or 


(118) 


Also  the  vessel,  instead  of  having  water  supplied  to  it  at 
the  velocity  of  the  vessel,  picks  it  up  from  a  body  of  water 
considered  at  rest  ;  thus  imparting  to  the  water  the  momentum 
Jlv,  requiring  the  work  per  second 

U2  =  Mv\  .     .     ....     .     (119) 

Hence  the  effective  work  done  by  a  jet  propeller  picking 
up  the  water  from  a  state  of  rest  will  be 

U=  U,  -  U2  =  M(v2  -v)v.   ....     (120) 

The  energy  exerted  by  the  pump  will  be  that  producing  the 
velocity  of  water  relative  to  the  earth,  or  %  M(v2  —  v)2,  plus^ 


HYDEAULIC    MOTORS. 


53 


that  doing  the  work  of  driving  the  vessel ;  hence,  the  energy 
expended  will  be 

i  M(v2  -  v)*  +  U; 

and  the  efficiency  will  be 


U 


(121) 


This  has  no  algebraic  maximum,  but  approaches  unity  as 
vy  the  velocity  of  the  vessel,  in  reference  to  the  earth,  ap- 
proaches v2  in  value,  the  velocity  of  the  jet  in  the  opposite 
direction  relative  to  the  orifice. 

If  V'2  =  v,  the  efficiency  will  fa  perfect  as  shown  by  (120),  but 
no  work  will  be  done  as  shown  by  (119).  This  would  be  the 
case  of  a  vessel  drawn  by  an  external  agency,  or  even  floating 
along  a  stream  ;  for  the  water  backward  relative  to  the  vessel 
would  equal  the  forward  velocity  of  the  vessel. 

The  mass  of  the  jet  per  second  will  vary  as  the  section  of 
the  orifice  and  velocity  of  the  jet ;  and  if  k  be  the  section  of  the 
jet,  then 

U=-kv2(v2-v)v,  •  .     ...."-.     (122) 

t/ 

hence  the  same  work  may  be  done  by  enlarging  the  section  &, 
and  properly  diminishing  the  velocity  v2  of  the  jet ;  but  as  v2 
is  diminished,  the  efficiency  is  increased,  as  shown  by  equation 
(121). 

If  v  =  10  feet  per  second  (about  6.8  miles  per  hour),  we  find 


»J 

u 

k 

E 

10 

0.*» 

00 

1.00 

15 

750  &' 

120.0 

0.80 

20 

2000  k  ' 

45.0 

0.67 

30 

6000  A' 

15.0 

0.50 

40 

12000  k' 

7.5 

0.40 

too 

90000  k  ' 

1.0 

0.16 

54  HYDRAULIC   MOTORS. 

The  sections  k  here  given  are  for  equal  works,  U.  If  the 
velocity  of  exit  be  constant,  then  will  the  work  increase 
directly  as  the  area  of  the  section  while  the  efficiency  remains 
the  same.  These  are  without  frictional  resistances. 

The  pressure  against  the  side  of  the  vessel  opposite  the 
orifice  due  to  the  reaction  of  the  water  will  be  found  from 
equation  (117)  by  dividing  the  work  done  by  the  space  over 
which  the  work  is  done,  or 


i\  =      =  MV,,  .  .  .  .  ;.  (123) 

which  is  the  momentum  of  the  jet  per  second  relative  to  the  orifice. 
To  impart  to  the  water  taken  up  the  uniform  velocity  v 
would  require  the  constant  pressure 

P.,  =  Mv  ; 
hence  the  resultant  pressure  producing  work  would  be 

P  =  P1-PZ  =  M  (v,  -  v),    .    '.    .     .     (124) 
and  the  resultant  work  would  be 

Cy=  M(v2  -  v)v,  .     .     .     .     .     .     (125) 

as  before  found  in  equation  (120). 

The  speed  of  a  jet  propeller  depends  upon  the  form  of  the 
vessel  and  the  nature  of  the  fluid  ;  but  the  pressures  due  to 
the  action  of  the  jet  will  be  the  same  whether  it  issue  into  a 
vacuum,  or  into  air  or  water,  or  a  more  viscous  fluid.  If  a  block 
be  placed  before  the  jet  so  close  to  the  vessel  as  to  obstruct  the 
flow  of  water  as  a  jet,  the  conditions  will  be  changed,  and  the 
forward  pressure  will  then  be  due  partly  to  the  direct  pressure 
exerted  by  the  pumps.  If  a  piston,  having  a  long  piston-rod 
projecting  against  a  firm  body  outside  the  vessel,  be  forced 
backward,  the  forward  pressure,  effective  in  driving  the  vessel, 
would  be  that  exerted  by  the  pumps  less  the  frictional  resist- 
ances. 


HYDRAULIC   MOTORS.  55 

The  efficiency  of  the  jet  propeller  as  a  motor  is  compar- 
atively small  in  practice.  This  is  due  to  the  great  loss  of 
energy  in  the  jet.  The  entire  energy  in  the  jet  is  lost.  If  the 
vessel  be  anchored,  and  the  velocity  of  the  jet  be  v2,  the  press- 
ure  will  be 

P, 
the  work  will  be 

Z7=0, 
and  the 

energy  lost  — 

If  the  speed  v  of  the  vessel  is  small,  then 
P!  =  Mv2, 

U  —  Mv2v,  nearly, 
energy  lost  —  %Mv22,  nearly, 

and  the  energy  lost  will  generally  exceed  considerably  the  use- 
ful work. 


5f)  HYDRAULIC   MOTORS. 

39.  The  difference  of  the  moment  of  the  momentum  of  the  water 
on  entering  and  leaving  the  wheel,  equals  the  moment  exerted  by  it  on 
the  ivheel.  (Proof  for  the  frictionless  turbine  by  J.  Lester 
Woodbridge,  graduate  of  Stevens  Institute,  1886.) 

Consider  the  effect  of  water  passing  along  a  smooth,  curved 
horizontal  tube  rotating  about  a  vertical  axis.  Conceive  the 
water  to  be  divided  into  an  infinite  number  of  filaments  by 


71 


71 


vanes  similar  to  those  of  the  wheel,  but  subjected  to  the  con- 
dition that,  at  each  point,  their  width,  id,  Figo  18,  measured  on 
the  arc,  whose  centre  is  0,  shall  subtend  at  the  centre  a 
constant  angle  dO.  Conceive  each  filament  to  be  divided  into 
small  prisms,  whose  bases  are  represented  by  the  shaded  areas 
db'c'd,  d'c'c'd"  and  abed,  by  vertical  planes  normal  to  the  vanes 
making  the  divisions  ae,  ef,  intercepted  on  the  radius  by  circles 
passing  through  the  consecutive  vertices  on  the  same  vane,  a', 
d',  d",  etc.,  equal. 


HYDRAULIC   MOTORS.  5T 

Let  p  —  the  radius  vector ; 

x  —  the  height  of  an  elementary  prism ; 
then,  dp  =  ae,  ef,  etc. ; 

pdOdp  =  abed,  etc.,  =  area  of  the  base  of  an  infinitesimal 

prism ; 

xpdOdp  =  volume  of  an  infinitesimal  prism ; 
xdpdOdp  =  m  —  the  mass  of  prism,  3  being  its  density 

or  mass  of  unit  volume  ; 

y  =  san  =  angle  between  the  normal  to  the  vane  at  any 
point,  and  the  radius  Oa  prolonged  through  that 
point ; 

V  =  velocity  of  a  particle  along  the  vane  at  p,  which  is 
assumed  to  be  the  same  in  all  the  vanes  at  the  same 
distance  from  the  centre  ; 

GO  =  the  uniform  angular  velocity  of  the  wheel,  and 
p  =  the  pressure  of  the  water  at  the  point  p  due  to  a 

head,  but  not  due  to  deflection. 

Let  p  be  the  independent  variable,  and  dt  the  time  required 
for  the  element  a'b'c'd  to  move  its  own  length,  dp,  and  ad  the 
distance  passed  through  by  this  element  circumferentially  in 
the  same  time,  dt,  then 

j*          dp 

at   =     ; j- 

v  sin  Y 
and, 

dt, 

aa  —  oopdt  =  cop-T-dp. 
dp 

The  mass  m  will  have  two  motions :  one  along  the  vane, 
the  other  with  the   wheel  perpendicular  to  the  radius.     By 
changing  its  position  successively  in  each  of  these  directions, 
both  its  velocity  with  the  wheel  and  its  velocity  along  the  vane 
may  suffer  changes  both  in  amount  and  direction,  as  follows : 
(I.)  By  moving  from  a  to  a',  Fig.  18,  in  the  arc  of  a  circle — 
(1.)  cop  may  be  increased  or  diminished  ; 
(2.)  cop  may  be  changed  in  direction  ; 


58  HYDBAULIC   MOTORS 

(3.)   v  may  be  increased  or  diminished ; 
(4.)  v  may  be  changed  in  direction. 
(II.)  By  moving  from  a  to  d'  along  the  vane — 
(5.)  oop  may  be  increased  or  diminished ; 
(6.)  cop  may  be  changed  in  direction  ; 
(7.)  v  may  be  increased  or  diminished ; 
(8.)  v  may  be  changed  in  direction. 

These  changes  give  rise  to  corresponding  reactions,  as 
follows : 

(No.  1.)  Since  the  element  is  to  move  from  a  to  a  in  the 
arc  of  a  circle,  oop  will  be  constant,  and  hence  the  reaction 
=  0. 

(No.  2.)  By  moving  from  a  to  a,  the  velocity  oop  is  changed 
in  direction  from  dk  to  a'k'  in  the  time  dt.  The  momentum 
is  moop,  and  the  rate  of  angular  change  is 

leak'   _  oodt  _ 

~dT   '~~dT     ^ 

and  hence  the  force  upon  the  element  producing  motion  in  the 
arc  of  a  circle  will  be  .radially  inward  and  the  reaction  will 
loe  mafp  radially  outward.  This  is  generally  called  the  centrif- 
ugal force,  as  designated  by  most  writers.  Resolving  into 
two  components,  we  have 

moo2p  sin  y  along  the  vane, 
moo2p  cos  y  normal  to  the  vane. 

(No.  3.)  According  to  the  conditions  imposed,  this  value  of 
v  is  the  same  at  a'  as  at  a,  hence,  for  this  case,  the  reaction 
will  be  zero. 

(No.  4.)  In  moving  from  a  to  a  the  velocity  along  the  vane, 
v,  is  changed  in  direction  from  at  to  at'  at  the  rate  GO  as  in 
No.  2. 

The  momentum  is  wtf,  and  the  force  will  be  mvw,  which 
acts  in  the  direction  In.  Since  the  particle  will  be  driven  by 


HYDRAULIC  MOTORS.  59 

the  vane  XY,  and  the  reaction  will  be   in  the  direction  rib\ 
which  being  resolved,  gives 

0  along  the  vane, 
—  mvoo  normal  to  the  vane. 

(No.  5.)  In  passing  from  a-  to  d'  ;  at  d'  the  circular  velocity 
will  be  greater  than  at  a  by  the  amount 


and  the  acceleration  will  be 

dp 

*-& 

requiring  a  force  vnwjr  tangentially  to  the  wheel  in  the  direc- 
tion of  motion,  the  reaction  of  which  will  be 

dp 


wv 

but  backwards,  and  its  components  will  be 

moo-jr  cos  y  along  the  vane, 

—  moo^r-sm  y  normal  to  the  vane. 

(No.  6.)  In  passing  from  a   to  d',  cop  will  be   changed  i 
direction  by  the  angle  between  k'a  and  k"d',  or 

a'od  _  a'y  -  dp  cot  y 

P  P- 

and  the  rate  of  angular  change  will  be 

cot  y  dp 


60  HYDKAULIC   MOTORS. 

and  the  momentum  being 

mcop, 


the  reaction  will  be 


dp 

moo  cot  y  ji> 


which  acts  radially  inward  and  its  components  are 

—  moo  cos  y  -si  along  the  vane, 

—  moo  cot  Y  cos  y  -j~  normal  to  the  vane. 

r/  v 

(No.  7.)  By  moving  from  a'  to  d\  v  will  be  increased  by  an 
amount 

dv  . 
-j-  dp, 
dp 

in  the  time  dt,  and  the  reaction  will  be 

dv  dp 

m  -j- .  -j,, 

dp    dt* 

which  will  be  outward  along  the  vane,  and  the  reaction  will  be 
directly  backward  along  the  vane,  and  hence  is 

dv   dp       ,         ,, 
—  m  -j-.  -j-      along  the  vane, 

0     normal  to  the  vane. 

(No.  8.)  In  passing  from  a'  to  d,  v  is  changed  in  direction 
by  two  amounts :  the  angle  y  changes  an  amount 


HYDEAULIC   MOTORS.  61 

This  is  negative,  for  a  differential  is  the  limiting  value  of 
the  second  state  minus  the  first,  and  the  first  is  here  larger. 

But  this  is  not  the  total  change,  since  y  "is  measured  from 
a  radius  making  an  angle 

dp  cot  y 

~^~' 

with  Oa'  as  in  No.  6  ;  hence  the  total  change  will  be  the  sum 
of  these,  and  the  rate  of  change  will  be  the  sum  divided  by  dt, 
which  result,  multiplied  by  the  momentum  mv9  will  give  the 
reaction,  which  will  be  normal  and  in  the  direction  l>'n  or 

0  along  the  vane, 

fcot  y   dp      dy   dp~\  ,  , 

mv       —  -  .  -T,  —  -T-  •  -T.   i  normal  to  the  vane. 

L    p       dt      dp    dtj 

This  completes  the  reactions.  Next  consider  the  pressure 
in  the  wheel.  The  intensity  of  the  pressure  on  the  two  sides 
ab  and  cd  differs  by  an  amount 


The  area  of  the  face  is  dc  x  x  =  xpdO  sin  y,  and  the  force  due 
to  the  difference  of  pressures  will  be 

xpdO  sin  y  ~-  dp. 

If  dp  is  positive,  which  will  be  the  case  when  the  pressure 
on  dc  exceeds  that  on  ab,  the  force  acts  backwards,  and  the 
preceding  expression  will  be  minus  along  the  vane.  In  regard 
to  the  pressure  normal  to  the  vane,  if  a  uniform  pressure  p 
existed  from  one  end  of  the  vane  V  IF  to  the  other,  the  resultant 
effect  would  be  zero,  since  the  pressure  in  one  direction  on  V  W 
would  equal  the  opposite  pressure  on  XT.  If,  however,  in 
passing  from  d  to  a,  the  pressure  increases  by  an  amount  —  dp, 


62 


HYDRAULIC   MOTORS. 


since  Va  is  longer  than  Xb,  the  pressure  on  Va  will  exceed  that 
on  Xb  by  an  amount 

—  dp .  x  x  ah  =  —  dp.x.pdO  cos  y  =  —  xp  cos  ydO  -f-dp. 
Collecting  these  several  reactions,  we  have 


NORMAL  TO  THE  VANE. 

(1.)        0 

(2.)   +  moo2p  cos  y. 

(4.)  —  moot. 

,„.  .dp 

(5.)  - 


(6.)   —  moo  cot  y  cos  y  -jr. 

(7.)        0 

\~coiy  dp    dy  dp~] 

(8.)   +  mv\  •  —  -  .  -JT—  —  .  JT  \. 
L    P      dt     dp    dt_\ 

(9.)  —  xp  cos  y  J-  dpdO. 


ALONG  THE  VANE. 


0 


+  ma>2p  sin  y. 


0. 

dp 
+  mcocosy-j-. 

dp 
—  mcocos  y-Ji» 


—  m 


dp   dv 
dt  '  dp' 


0. 


ap  ,    ,„ 
—  xp  sin  y  -j-  dpdO 


The  sum  of  the  quantities  in  the  second  column,  neglecting 
friction,  will  be  zero  ;  hence 

motfp  sin  y  —  m  -rr  .  -, xp  sin  y  -j-  dpdO  =  0.  .     (123) 

Substituting 

dp          .  ,        7/3 ,        m 

—r.  =  v  sin  y,  and  xpdudp  =  ~^ 

and  dividing  by  m  sin  y,  we  have 


co2pdp  —  -  dp  =  vdv. 


(124) 


HYDRAULIC   MOTORS.  63 

Integrating, 

r  ~|  limit         r       -I  limit 

Uc»V--P  =U^2         ....     (125) 

L-  $J  limit         L_       -I  limit 

The  sum  of  the  quantities  in  the  first  column  gives  the 
pressure  normal  to  the  vane,  which,  multiplied  by  p  sin  y,  gives 
the  moment.  This  done,  we  have 

fP  r»\  .       dy 

ojy    —  GO  cos  y  —  2  )  —  pv  sin  y  -j— 
\v  /  dp 

=  mv  sin  p  - 

cos  y  dp 

4-  v  cos  y  —  p ^-  -r~ 

vo     dp 

fi  O 
Putting  mv  sin  y  =  o—   dpdft,  where  Q  is  the  quantity  of  water 


flowing  through  the  wheel  per  second,  and  integrating  in  refer- 
ence to  6  between  0  and  2?r,  we  have 

dM<=M  \o*P  (^  cos  y-  2]  -pv*m  y^+v^y-p^  ^1  dp 
L      \v  dp  vd  dj 


g  L      \v  dp  vd  dp 

Multiplying  (124)  by 


P 

—  cos  y, 

v 


we  have 


>2p2  7        p  cos  v  dio  7  dv  7 

—  cos  yelp  —  L  -  L  -/  dp  =  pcos  y  —.  dp, 
v  v6     dp  dp 


which  substituted  above  gives 
d  J/t—  _  ^~— 


dp 
the  integral  of  which  is 

J/;  =  tf  Q  [—  G?p2  +  p^  cos  y]  -r-  g 


^  Cos  r]  !iSJt:     .      .      .     (127) 

<7 


64:  HYDRAULIC    MOTORS. 

But  GO  p  —  v  cos  y  is  the  circumferential  velocity  in  space  of 
the  water  at  any  point,  and  S  Q  p  [GO  p  —  v  cos  y]  is  the  moment 
of  the  momentum  ;  hence,  integrating  between  limits  for  inner 
and  outer  rims,  the  moment  exerted  l>y  the  water  on  the  wheel 
equals  the  difference  in  its  moment  of  momentum  on  entering 
and  leaving  wheel. 

Let  the  values  of  the  variables  at  the  entrance  of  the  wheel 
be  pn  Xi,  ^,,^i,  and  at  exit  pn  y»  v,,  p.2. 

Equations  (125)  and  (127)  become 


M  =  <J  Q  [GO  (Pl2  -  p2<)  -  Pl  Vl  cos  y,  +  P2  v,  cos  y^.    (129) 

U  =  M,co  =J*  GO  [GO  (p*—pf)  —  pi  Vi  cos  y,  +  pa  vf.  cos  y^  (130) 
y 

Equation  (58)  for  a  frictionless  wheel  in  which  /^1  —  //2  =  0,  re- 
duces to  equation  (130).  This  principle  simplifies  the  solution 
of  certain  special  cases.  Thus,  in  the  Barker  Mill,  page  44,  the 
momentum  of  the  water  entering  the  wheel  will  be  zero,  but  of 
exit  will  be 


where  V^  is  the  velocity  of  exit,  relative  to  the  earth,  perpendicu- 
lar to  the  arm,  and  the  moment  will  be 


.-.  U  =  M  V,  •  r,  v 
=  Ps, 

where  P  is  the  effective  pressure  on  the  arm  opposite  the  orifice 
of  the  jet,  and  s  the  space  passed  over  by  the  orifice  in  a  second 
of  time.  If  jPj-  be  the  pressure  on  the  arm  due  to  the  reaction, 
M  V»  of  the  jet,  and  P^  the  pressure  which  imparts  to  the 
water  a  momentum  MI\GO,  then  P  =  P:  —  P9. 


HYDRAULIC    MOTORS  65 

But 

F  =  v,  —  r,  GJ  ; 

M  =  —  -  ^  ^  *'• 
g  g 

(1  +  /!)«.•  =  2  j-  H  +  a>"  r,1 ; 

.-.  ?7  =  — il'  (v,  -  r,  co)  r,  <a  (132) 


as  in  equation  (105). 

40.  Again,  if  the  water  quits  the  wheel  radially,  then   the 
moment  of  the  momentum  of  the  quitting  water  will  be  zero,  and 

U  =  M  V  ri\  cos  a  •  co. 
But 

V  cos  of  =  F~t, 

the  tangential  component  of  the  velocity,  or  velocity  of  whirl ; 

.-.   U  =  M  Ft  r,  oo.  (133) 

41.  In  the  frictionless  lianklne  wheel  the  velocity  of  whirl 
equals  the  velocity  of  the  initial  rim  of  the  wheel. 

.-.  K***<*i 

.-.  U  =  Mr*  oo\  (134) 

The  work  will  also  equal  the  potential  energy  of  the  water, 
W  H  =  d  Q  H,  less  the  kinetic  energy  of  the  quitting  water, 
£  M  V*  (less  the  energy  lost  in  resistances,  //  v^  which  in  this 
case  we  neglect) ; 

.-.  U=8  QH-$M  F22, 

and  since  the  water  is  assumed  to  quit  radially 

F2  —  7'2  co  •  tan  y^  —  rl  GO  tan  a.  (135) 


66  HYDRAULIC    MOTORS. 

The  three  preceding  equations  give 


/     Ig  II 

1\   <X>—A/-    -  j 

\     2  +  tan*  a 
as  in  equation  (76). 

42.  Again,  if  the  crowns  are  parallel  discs  and  the  initial 
element  of  the  bucket  is  radial,  and  if  the  water  quits  the  wheel 
radially,  and  if  the  velocity  of  whirl  equals  the  velocity  of  the 
initial  rim,  we  have 

U=  Mr:  «?,  (136) 

as  in  equation  (134).  But  y^  will  not  be  the  same  as  in  (135). 
To  find  it  we  have,  neglecting  the  thickness  of  the  walls  of  the 
buckets, 

2  7t  >\  vl  =  2  7t  i\  sin  y^  -  v^ 

V^  —  V  sin  a 
rt  GO  =  V  cos  a 
F"2  —  t\  oj  tan  /2 
v,  =  i\  &  ; 

/\a  tan  OL 
/.  tan  y,  =:   -  -  ;  (137) 

Vr*  ~  /Ji4  ian*  a 

.\U=  d  QII-^M  V; 

?\*  ta?i*  a 

l  M  r?  v*  -  --  -—  7-.  (138) 

/•*  —  r*  tan3  a 


Equations  (136)  and  (138)  give 


34  —  r*  tan9  a  (139) 


HYDRAULIC    MOTORS.  67 

Practical    and    Experimental    Data    and    Results. 
A  STUDY  OF  THE  TEEMOXT  TURBINE. 

(Revision  of  a  paper  by  the  Author  in  Vol.  XVI.  of  Trans,  of  Am.  Soc.  of 

Mech.  Engineers.) 

44.  The  Tremont  turbine  furnishes  an  excellent  example  for 
testing  the  theoretical  formulas  for  the  proportions  and  deliver- 
ance of  turbines.  The  one  here  analyzed  was  made  by  J.  B. 
Francis,  engineer,  after  the  general  pattern  of  the  celebrated 
U.  A.  Boyden,  Esq.,  turbines,  which  yielded,  from  careful  ex- 
periments, an  admitted  efficiency  of  88  per  cent.  The  dimen- 
sions of  a  Tremont  turbine  and  the  careful  and  somewhat  ex- 
haustive efficiency  tests  are  fully  set  forth  by  Francis,  in  his 
work  entitled  Lowell  Hydraulic  Experiments.  (Fig.  19.) 

The  workmanship  on  these  wheels  was  of  high  grade.  The 
crowns,  which  were  of  cast  iron,  were  accurately  turned  in  a 
lathe,  and  the  partitions  (or  walls)  of  the  buckets  were  of  Russia 
sheet-iron  plates,  ^9¥  of  an  inch  thick.  These  plates  fitted  into 
grooves  carefully  cut  in  the  crowns,  on  which  were  tongues  pro- 
jecting through  mortices  in  the  crowns,  and  the  former  headed 
down,  thus  securing  the  crowns  to  each  other  without  bolts  or 
rods,  and  forming  smooth,  unobstructed  passages  for  the  flow  of 
water.  In  the  Boyden  wheel  a  "diffuser"  was  added,  which 
consisted  of  two  crown -like  pieces  outside  the  wheel,  but  not 
rotating  with  it,  the  space  between  which  was  diverging,  produc- 
ing a  diminution  of  velocity  of  the  escaping  water,  causing  more 
of  its  energy  to  be  imparted  to  the  wheel.  This  device,  it  is 
said,  added  some  2  or  3  per  cent,  to  the  efficiency  of  the  wheel. 
The  Tremont  turbine  was  not  provided  with  this  device.  (Fig.  30.) 

We  have  selected  for  analysis  the  experiment  by  Francis 
which  yielded  the  highest  efficiency ;  for  our  turbine  formulas 
are  strictly  applicable  only  when  the  wheel  is  so  proportioned  and 
run  as  to  give  a  maximum  efficiency. 


68 


HYDRAULIC    MOTORS. 


•auBA  am  ^0. 

pajBOipm  SB  paqAv  aqj  Sui 
-ABai  iajBAY  aqi  jo  uoijoajtQ; 


C050         OSt-OGO 


t-OGOO         t-  0         ^COO 

roo<T-oj      cor-i      i^  1-1  <M 


00  C5        O5  »C  T-I  CD  GO        O  OO        ~  »  CS 
T-KM        WCO^^TlH        ICO        OXCi 


oj  aup  XjpopA  aqj  oa  paq.vv 
aqj  joaouaaajuuio.no  jouajui 
aqj  jo  Xipo[»A  aqj'  jo  OIJBJI 


•puooas  aad  jaaj  3 
ui  paqA\  aqj  jo  aouaaajuino  ^ 
-a'p  aouajui  aqj 


THi-lOOi-iOOOOOi-iOOT-iOOO 


0  t  0  T-  ^  CC  CO  r-  0*  CO  1C  t- 
CO  CO  C5  "*  CI  C7  CO  CO  ^  i.t  00  C1?  CO 


~<  O  cs  c:  r^ 

«  «   r-   rH   50 


GO  00  t-  l>  CO  0 


•puooas 

J2     Ja'l  laaj  ui  paqAv  aqj  uo  Sin    ^ 
-JOB  i[t:j  aqj  oj  anp  XjjoopA 


cox-^co^co^rcooci 
^5  tc  S  co  co  t^  t-  <--~  t-  x 

X'  X'  X'  X  X  X  X  X  X  GO  X  X  X  X  X  GO  x' 
CQ  O">  OJ  Ci  C3  C"<  (?>?  d  O1  C 


"popuadxa  ja.vvod  am 


t—  1C  C5  ^         IC-^fC^O'^t1         OO         -^fCOO 

^HO'—?-      TtHco^oco       xco      oicSco 

t^  CO  t^  CO        CO  O  t-  X  X        X  S5        CI  X-  GO 


•puooas  ,iad  looj  auo 
S  pasuu  aiodnpaioAW  npnuod  ui     9 
aajBAV  am    jo  aaAvod 


^^f.-c--rtioxo>^j.-rcixo'*cocs 

l>  CO  CO  O>  X  O  '-fi  CO  CO  L~  i.-O  T-  CO  X  CO  CO 

1-1  x  co  LT  07  —  x  j.t-  cocoaoe«»c5^<TH 


x  j.t-  cocoaoe«»c5^<TH 


•puooas  jad  jaai 

£     oiqno  ui   '.naA\  aqi^  passml    O1 
iaiBAv   jo 


0  07  07  ^  CO  0  07  t-  CO  X  ^  T"  «  CO  X  CJ  CO 

t^  x  ^  co  1-1  x  co  o  co  r—  TT  a  ci  x  co  co  i— 
^p  co  ^  J--  co  —  t-  co  0  co  c:  O  X  c?  co  X  cs 

OCQMX^OOS-^tOCOCOO-^COOJOi;- 
CO  C5  t>  "^  CO  O?  I-H  O  O  OS  '-'  C5  X  Cl  GO  GO  t^ 


aqi 


-jaaj  in 


^COOOOCOOOCOXClClCOOOi-'O 
1O  O  O  O  T-H  LO  CI  X  X  CI  O  C5  O  CO  T-  "t  CO 

03  oi  oj  o?  o?  03  oi  07  oi  oi  oi  oi  oi  oi  oi  o?  o> 


•puooas 

0    aad  jooj  auo  pasiBa  siodnp 

1-1    -JIOAB  apunod  ui  a^Baq  aqj  jo 

uoijouj  aqj  JO  'j 


CO  CO  IO  CD 


SS 


Ci  C5        X  X 


X  C-          C7  "^  CO 


XX 


•puooas  .iad 


aqj  jo  suoi}ii[OAaa  jo  jaqumx; 


,o 


OOXXCOCi^t(^CtXXOX"cOXrf 

I  ^J  -4  ,-i  T_;  r-i  o  o  o  o  ~  o"  o  T-*  o'  o"  o 


•saqoui  ni  aiB§ 
SnilBinSaa  aqi' 


^H     -juauiuadxa  am 


HYDRAULIC    MOTORS. 


69 


O  O7  l>  O7  r-  O  O7  GO  O  O7  OS  O 

CO  CO  TH    07  O  O  07  CO  CO    xH 


»o  TH  TH  t~  ^eswcftGOJGOTHOoiTHt-^THOicttt-ocososocfcsoo&o 

^r  O  O  O  IQ  r^  ^  T»<  30  O  6«  CO  T*  OS  00  CO  CO  C*  00  «D  O»  «>  O6  0»  -3  SO  CO  SO  N 

t>  CO  TH  CS  CO  —  i  CS  O  Cl  TH  O  OO  CO  OO  TH  L-  c-  tO  TH  CS  O  CS  O7  O?  Xi  O  CS  GO  CO 

TT  CO  TH  CO  CO  CO  to  tO  O  00  t-  O  SO  tO  1C  O7  30  30  GO  t-  t-  TJH  CO  O  to  OS  ?C  •*$  CO 


09 

70  TH 

—  :OOSO7O7X>C-t>OSTH 
OOX>THO3O3COTtltOO5 

OS 

T 

Q  T-I  cs  CS  -H  JO  TH  0 
-i  07  CO  ^  TH  0  f 

^iS 

O7  TH  to  O7 
§*  f-  CO  00 


»^4OQOiH   «O  OS  CO 


CO  00  «O  TH  O7  O  TT  O*  t-  O  TH  IO  C^  CO  O  O  CS  CO  O  CO  CO  TH  O  O  O?  GO  O  CO  I-  CS 
O?  00  T-I  TH  CO  O  O  T-I  O7  CO  1C  O  O  CO  OJ  •-:  CS  GO  CO  IO  X>  Oi  CO  OO  O  Tf  O  CO  O  >-H 
t-  T~i  CO  O  O  CO  ^f1  CQ  t--  L-  CO  O  O  Tt<  00  O  OS  c;  ^p  O  «O  T-I  t-  T  CO  C<«  QO  T-H  O  O> 


"   -   OS  OO  t- 


A  »o  i  --  1  O  CT  O  OS  CO  O  O  T-I  GO  CO  O  OO  ^  O  C*  OT     -  ^  t-  r-i  O  C5 

!?07DClCJO>?OiOO^r-iOi>i-i^^OOOTHCl^t-t-C30'-(Oi.':"^-O?GOCiOCO 
00  C?  O)  70  O  CO  00  -X)  QO  O  .00  GO  OO  OS  O5  OS  O  T-H  O  i—  T—  C^J  C-7  CO  i>>  ;30  CO  CO  "^_  O  -^  OS  O  O 

GO  ;r  06  a5  -06  od  ad  a5  TD  ad  06  ad  06  06  a5  od  a>  cs'  os'  ci  ri  cs'  os  os  06  a5  cs  os"  os"  cs'  as'  as'  o  o 

OlC^Ol^l  0>  <?>  ^<  07  O?  O?  07  Oi  07  07  07  O<  O7  O7  O7  O<  O7  O7  O7  O7  O7  O7  O7  O7  O7  O7  O7  O7  CO  CO 


O  O7  •»  O  GO  t^  TP  O 
T-t  OS  t^  O  •*  rf  O  C-  O7 
OS  00  ~X>  •*#  t-  O 


O  O7  O7  <^  O  O  O  T-H  O  OS  I-  rt< 

O  O  -^  t-  rH  CO  O7  CO  O  Cl  T-H  O 

CO  O  -H  ^r1  X)  7D  CO  O  O7  -rf  O  -f 

«D«C5«oiot^t^  co  ^  -^  ^—  T-I  07 


O  03  ^H  30  CO  O  10  l>  O  Z>  07  iO  CO  O  i--  O_  IT  OS  ^  T-H  as  O  00  O  O  TH  CO  O  O  'JO  O  O 
t^  O  O  "*  OS*  O7  CO  O  t^  Tjn  O7"  O7  O  «3  O7  ^'  OS*  ^  00  C^  O  O  O  CO  OS  GO  Tf'  O7' 
t^  O  YJ  CO  O  CO  T-I  CO  TH  O  O  30  rf  O  i-T  OS  C5  O  OO  O  O7  T-  3D  t-  J>  T-t  O7  ^ 


CO  T 
70 


O  O7  O  O7  CO  O  T-H  O  C5  ~H  t-  I-  —  '  CO  OS  C5 


L*>TH:o:O:OL-~THOCSCSGOOTHTHOO7OlOTHC5t~O7  — lOGOCSTHCOOGOCOOCSCO 
^  2  j ,  /3  jo  CO  TH  OS  O  TH  —  07  t-  TH  O  CO  t-  GO  -H  cs  CS  «0  TH  TH  rH  O  CO  — '  10  10  OS  T-,  CS  CO 

-H  os  as  10  to  o*  O7  o  to  co  to  o  as  o  as  as  TH  o  to  to  -^  TH  co  TH  o  TH  to  07  c~  — i  to  07  o  t- 

T-,  O  t--  i  -  O  O  O  TH  TH  CO  t-  O  O  CO  OS  GO  to  OS  to  O7  IO  O  tO  IO  CO  tO  TH  «O  GO  00  CO  O7_  to  TH 
jjj  ^5  ^j5  TO"  to'  O7*  GO*  GO*  GO  CO'  t*-'  J>"  io"  CO'  -H'O  OS*  T-H  GO*  TH  TH  GO'  O  O  l^  O  O  O  TH  t>  O  GO"  GO*  l> 

CO iOCOCOCOCOCOCOCOTfCOCOCOCOCOCOTHO>THTHT-iOOOCOCOaOt~t-OOCOCOCO 


t—  J>  OS  C-  O  •***  O  iO  O  »C  O 

os  as  t'*  ^  as  as  as  t~~  as  as  as 

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70  HYDRAULIC    MOTORS. 

The  coefficients  /*,  and  /*2  are  uncertain  quantities.  Weisbach 
considers  /<,  =  0.10,  and  /*2  from  .05  to  0.10.  We  will  assume 
jul  —  0.10,  and  for  the  assumed  Boyden  wheel  yw2  =  0.05.  Our 
analysis  will  show  that  /^Q  =  0.15  nearly,  for  the  Francis  wheel. 

For  the  wheel  we  now  analyze  we  have  (see    Lowell  Hyd. 


^  =  3.375  feet  ;  r,  —  4.146  feet. 

a  -  -  21°  }  as  measured  from  Plate  III.  of  Experiments* 
y^  =  10°  >  (We  have  used   ;/a  =  13°   and   15°   for  reasons 
Yi  =  90°  )  which  appear  later.) 

Q  =  138.2  cubic  feet. 

//  =  12.9  feet. 

A,  =  15.11  feet. 

A,  =  2.2  feet. 

U'  =  111,218.1,  total  power  of  the  water  in  foot-pounds 

per  second. 

E  —  0.79375  efficiency. 
H.P.  —  161,  about,  as  measured  by  the  brake. 

n  —  0.85106,  number  of  revolutions  per  second. 

y^  =  0.9368  feet,  depth  between  the  inner  edges  of  the 

crowns. 
y2  —  0.9314  feet,  depth  between  the  outer  edges  of  the 

crowns. 

JV  —  44,  number  of  buckets. 
g  =  32.16. 
With  this  data  we  solve  as  follows  : 

I2ev.  per  sec.  —  --  .  (1^0) 

2    71 

*  Rodmer  in  his  work  gives  a  =  28°,  y\  •=  90°,  y2  =  22°,  but  these  do  not 
agree  with  the  plates  given  by  Francis,  neither  do  they  produce  results  agree- 
ing with  Experiments.  Rodmer  considers  these  as  mean  angles  of  the  stream. 


FIG.  19. 


HYDRAULIC    MOTORS. 

TABLE  IX. 


71 


a  =      21° 
V2  =     13° 
j*a  =  0.20 

a=      21* 
Y2  =     13° 
im2  =  0.10 

a=      21" 
V2  =      13° 
M2  =  0.05 

Measured 
Values. 

w  Eq  (16)  

5  369 

5  476 

5  514 

Rev  per  second. 

0  855 

0  872 

0  879 

0  351 

Emu  Per  cent.  Eq.  (16a) 
vt  feet.  Eq.  (18)  
Vi  feet  Eq  (19)  

79.308 
6.968 
23  115 

84.84 
7.102 
23.989 

87.76 
7.157 
24  525 

79.375 

^2  fc\  T\  /r»n\ 

-  =  T  =  Eq.  (20)  

Vi  KI 

26  feet.  Eq.  (21)  
yi  feet  Eq  (21)  

3.316 

0.957 
1.103 

3.37 

0.939 
1.063 

3.426 

0.932 
1  039 

0.9368 
0.9314 

It  will  be  observed  that  for  /*2  =  0.05  the  efficiency  is  over  87 
per  cent.,  and  if  a  diffuser  would  add  2  per  cent.,  we  would  have 
an  efficiency  exceeding  that  admitted  for  the  Boyden  wheel. 
But  it  seems  improbable  that  the  prejudicial  resistance  can 
be  so  low ;  and  it  is  well  to  observe  that  for  a  =  20  degrees, 
ftj  =  0.10  =  jw2,  and  y^  —  10  degrees,  a  theoretical  efficiency 
nearer  90  per  cent,  will  be  found. 

It  will  be  seen  that  for  //2  =  0.20,  some  of  the  numbers  given 
}n  the  first  column  of  the  preceding  table  agree  well  with  actual 
values.  Thus  the  revolutions  are  : 

Computed     .     ....     .     .     ,     .     51.3  per  minute. 

Measured -,     .     .     51.1  per  minute. 

Also  the  efficiency  is  : 

Computed    .. 79.308  per  cent. 

From  measurements  .     .     ,     •     •     *     79.375  per  cent. 

The  initial  depth  between  the  crowns  is : 

Computed     ....     0     ....       0.957  feet. 
Measured  0.9368  feet. 


Difference 


.0202  feet. 


72  HYDRAULIC    MOTORS. 

The  outer  depths  between  the  crowns  are  : 

Computed 1.103  feet. 

Measured      .     .  .9314  feet. 


Difference ;    .     .     .         .1716  feet. 

or  .     .     ...    ..     .     .     .       2.059  inches. 

Not  only  do  the  computed  depths  exceed  the  measured  ones, 
but  our  computed  depth  is  less  for  the  inner  rim  than  for  the 
outer,  while  the  measured  values  are  the  reverse.  This  cannot 
be  dismissed  with  the  remark  that  the  wheel  was  improperly 
proportioned;  for  the  138.2  cubic  feet  per  second  passed  the 
weir  ;  therefore  this  requires  further  investigation. 

45.  The  depths  between  the  crowns  were  determined  as  follows  : 
Dropping  the  subscripts  in  the  notation  and  placing  p  for  ?\  or  ?'2, 
we  have 

y  (2  7t  p  —  Nt  -4-   sin  y)  v  sin  y  =  Q  (1^1) 

in  which  2  n  p  =  whole  circumference  at  distance  p  from  centre, 

JV7  =  thickness  of  N  bucket  plates. 

^Yt  -r-  sin  y  —  thickness  measured  on  an  arc. 

y  (2  n  p  —  Nt  -r-  sin  y)  =  free  ring  section  for  passage  of 
water. 

v  sin  y  =  radial  component  of  the  velocity  in  the  bucket,  and 
the  product  gives  the  quantity  of  flow  in  feet  per  second,  or  Q. 
Hence,  for  initial  and  terminable  values,  we  have,  since  Q  is 
constant, 

y._^8,r>nr.-37)_*-(8»x8-M-££ir) 

y.       v~  (2  7t  i\  sin  y.  —  JV?)  /„  44  X  9 

*'1  *       ^  *  *      *  /  At        I     VI     Tf    'i»      OT  11     /*  * 


If  the  thickness  of  the  bucket  plates  be  neglected,  we  have 
simply  (yl  being  90  degrees), 
?/., ?*, 


HYDRAULIC    MOTORS. 


73 


which  for 


=  13  degrees  gives 
3.38  6.968 


23.115  X  0.2+5  -- 
or  the  depths  would  be  equal  ;  and  that  the  outer  depth  shall  be 
less  than  the  inner,  y^  must  exceed  13  degrees.  We  tried  15 
degrees  and  made  other  computations,  as  follows  : 


TABLE  X. 


0  =  138.2 
a  =  21° 
72  =  15° 
/ita  =  0.20 

0  =  K35.5 

a  =  21° 
Y2  =  15' 
M2  =  0.15 

q  =  138.2 

a  =  23° 
y2  =  14- 
M2  =  0.20 

Q  =  138.2 
a  =  23° 
V2  =  15° 
M2  =  0.20 

Measured  . 

(t)        .  .           . 

5  339 

5  407 

5  364 

5  346 

Rev.  per  second 

0  853 

0  867 

0  854 

0  851 

851 

Emax.  per  cent  
Vi  feet      

78.33 
6  954 

80.9 

7  017 

78.90 
7  696 

78.10 
7  67 

79.375 

V?  leet  

23.048 

23  573 

23  15 

23  19 

y\  feet 

0  9439 

0  9317 

0  862 

0  870 

9368 

y2  feet     ... 

09346 

0  9104 

1  00 

0  945 

9314 

In  the  first  column  the  buckets  are  still  too  deep  and  the 
efficiency  too  small.  Since  the  gate  does  not  fit  water-tight,  a 
part  of  the  138.2  cubic  feet  which  passed  over  the  weir  may  have 
escaped  at  the  gate,  and  hence  the  entire  quantity  may  not  have 
done  work  on  the  wheel.  If  there  be  4-  of  an  inch  clearance  at 

o 

the  gate  (this  assumption  is  gratuitous,  but  will  serve  for  illustra- 
tion) there  will  be  an  annular  opening  of  2  X  3-J-  X  3.38  X  -J-  X 
rV  =  0.222  square  feet.  It  will  be  found  hereafter  that  the  in- 
ternal pressure  at  the  gate  is  2648  pounds  per  square  foot,  and 
the  atmosphere  and  2.2  feet  head  in  the  wheel  pit  gives  an  out- 
side pressure  of  2254  pounds,  leaving  394  pounds  inside  effective 
pressure  at  that  point,  which  would  produce 

a  velocity  =  |/2  g  X  |—  —  20  feet  nearly ; 

62.2 

hence  the  volume  of  water  which  would  escape  under  these  con- 


74  HYDRAULIC    MOTORS. 

ditions  would  be  0.62  X  0.222  X  20  =  2.75  cubic  feet  nearly, 
allowing  0.62  for  the  coefficient  of  discharge.  This  would  leave 
135.5  cubic  feet  of  water  to  do  work  on  the  wheel.  The  leakage 
produces  a  slight  discontinuity. 

The  computed  (hydraulic)  efficiency  ought  to  equal  the  brake 
efficiency  plus  that  of  the  shaft  friction  plus  that  lost  by  leakage  ; 
so  another  computation  was  made,  using 

Q  =  135.5,  a  =  21°,  y,  =  15°,  ^  =  0.10,  ^  =  0.15, 

the  results  of  which  are  in  the  second  column  of  Table  X.,  and 
are  very  good.  The  revolutions  are  one  more  per  minute  than 
those  observed  ;  the  efficiency,  1.525  per  cent,  greater  than  that 
observed,  which  difference  is  probably  about  right ;  the  inner 
depth,  2/j,  is  almost  exact,  but  y^  is  perceptably  less  than  that 
measured.  A  reduction  of  2  per  cent,  of  the  speed  to  reduce  it 
to  that  observed  would  reduce  the  efficiencies  but  slightly,  as 
shown  by  the  experiments  on  page  68.  Thus,  in  experiment  32 
the  speed  is  about  2  per  cent,  less  than  in  No.  30,  which  gave 
the  maximum,  but  the  efficiency  is  reduced  from  0.79375  only 
to  0.79294.  In  No.  48  the  speed  is  nearly  6  per  cent,  less,  but 
the  efficiency  is  reduced  about  0.7  per  cent.,  or  to  0.789.  It  will 
be  observed  that  the  wheel  delivers  nearly  the  maximum  work 
when  the  speed  is  within  5  or  6  per  cent,  above  or  below  that 
which  gives  the  maximum  efficiency. 

There  is  some  uncertainty  about   the  correct  measure  of  the 

section  of  discharge.     In  equation  (141) gives  the  thick- 
sin  7 

ness  of  a  bucket- wall  measured  on  the  arc  whose  radius  is  p,  on 
the  hypothesis  that  the  arc  is  a  straight  line,  which  is  sufficiently 
accurate  when  the  wall  is  very  thin  as  in  this  case.  Also  v  sin  y 
is  assumed  to  be  constant  for  all  points  in  a  ring  section  which  is 
accurate  for  a  fillet,  and  sufficiently  so  for  a  very  narrow  stream. 
But  these  conditions  are  changed  when  applied  to  finite  streams, 
as  in  actual  wheels.  This  will  be  more  clearly  seen  if  equation 


HYDRAULIC    MOTORS.  75 

(141)  is  so  transformed  as  to  apply  to  a  single  bucket,  for  which 
it  becomes  at  the  terminal  end 


ft 


in  which        /2  =  qf,  Fig.  20,  the  distance  measured   on  the   arc 
covered  by  one  bucket  and  partition-wall. 

t  =  he,  the  thickness  of  the  partition-wall. 

Francis  assumed  that  the  correct  section  of  discharge  was  not 
the  arc  ad,  but  was  the  least  section,  and  experiments  and  com- 
putations combined  confirm,  at  least  approximately,  the  correct- 
ness of  this  assumption.  Hence  with  a  as  a  centre  find  by  trial 
an  arc  which  will  be  tangent  at  h — draw  ah,  it  will  be  normal  to 
dh,  and  will  be  the  base  of  the  least  section  of  fiow.  Through 
the  middle  point  of  ah  at  cj  pass  the  middle  arc  gt  of  the  bucket. 
Then  considering  g  as  the  terminus  of  the  middle  fillet,  find  7, 
r,  and  v  for  that  point,  and  call  that  velocity  the  mean,  and  let 
^m  be  the  mean  velocity  ;  then  vm  ah  .  y  .  N  =  Q.  In  this  case 
y  at  g  will  exceed  y.2  at  «,  r  at  g  will  be  less  than  r9  at  a  and  vm 
will  be  less  than /y2.  We  have  not  determined  these  quantities  at 
g,  for  our  knowledge  of  what  takes  place  in  the  section  ah  is  riot 
definite,  for  it  is  a  state  bordering  on  discontinuity,  since  the 
stream  follows  the  bucket  to  that  section,  and  from  that  flies  off 
into  space  in  the  direction  nahm  more  or  less  radial.  We  have 
not  positive  means  of  determining  whether  this  view  is  correct, 
for  we  do  not  know  if  the  bucket  was  filled  with  a  live  stream 
from  end  to  end  ;  but  we  assume  that  it  was,  and  are  seeking 
conditions  which  will  be  consistent  with  this  assumption.  We 
have  already  seen  that  by  increasing  yt  from  13°,  the  vane  angle, 
to  15°  gives  results  which  agree  closely  with  our  assumption,  and 
we  anticipate  that  where  the  terminal  angle  yt  is  small  and  the 
buckets  numerous  a  similar  result  will  follow  in  other  cases. 


76  HYDRAULIC    MOTORS. 

Prolong  ah  to  <?,  and  make  cq  and  hi  perpendicular  to  ah  •  aj 
tangent  at  a  intersecting  cq  ztj  ;  then 

aj  sin  ajc  =  ac, 

ah  =  ac  —  t  (very  nearly). 

Drawing  db'  tangent  to  the  arc  of  the  bucket  at  a,  then  Vaj  = 
Yv  and  ajc  will  exceed  y^  while  aj  will  be  less  than  the  arc  af, 
so  that  it  may  require  a  computation  to  determine  whether  of 
sin  Y*  —  t  is  greater  or  less  than  the  least  measured  distance,  ah. 
In  Fig.  20,  y.  is  25° ;  in  the  Tremont,  10°.  The  computed  dis- 
tance will  be 

I  =  g-5  *  4  sin  10°  -  -  _J? =  0.1095  feet. 

44  12  X  64 

The  least  measured  distance  ah  =  0.1875  feet.  The  com- 
puted velocity,  -ya,  will  exceed  the  mean  velocity,  and  Z>,  being 
less  than  the  least  base,  there  is  a  partial  compensation  in  the 
product  b  v2,  so  that  y2  =  Q  -T-  2? b  v^  approximates  toward  the 
correct  value.  According  to  our  analysis, 

y,  v,  (of  sin  13°  -  t)  =  J. 

gave  too  large  a  value  for  y2 ;  then  we  found  by  trial  that 
y2  v,  (of  sin  15°    -  t)  =  j£ 

gave  a  value  for  y2  a  little  too  small.    Had  the  mean  velocity,  vw  ' 
been  found,  it  would  have  been  necessary  to  increase  y  still  more? 
that  we  may  have  the  equality 

y2  vm  (af  sin  y  —  t}  =  j^. 

But  in  this  case  y2  would  be  a  mean  depth  along  ah.  If  the 
bucket  be  divided  into  very  small  imaginary  ones,  say  n  such, 
we  have 


FIG.  20. 


HYDRAULIC    MOTORS.  77 

ad    .  O 

y*  v*  — sm  r2  =  -4^ 

n  nN 

but  this  is  equivalent  to  assuming  that  the  velocity  is  uniform 
throughout  the  cross-section  and  equal  to  va,  which  is  true  only 
for  very  narrow  streams.  Cancelling  n  will  reduce  the  equation 
to  the  same  as  (141). 

From  the  results  in  Tables  IX.  and  X.,  we  conclude  that  the 
prejudicial  resistance  ^  is  less  than  0.15  and  exceeds  0.10  in  this 
wheel.  The  terminal  angle  of  the  guide  a  perhaps  ought  to  be 
increased  in  the  analysis,  for  the  same  reason  as  that  applied  to 
Yv  but  not  to  the  same  amount,  for  the  pressure  at  the  gate  will 
cause  the  water  to  follow  the  back  of  the  guide  more  nearly  to 
its'  end. 

We  try  another  computation,  making 

Q  =  136,  a  =  22°,  y,  =  15,  ^  =  /i,  =  0.13, 

with  the  following  results : 

Hev.  =  0.862 ;  E  =  0.821 ;  yl  =  0.884,  y,  =  1.038,  which  re- 
sults are  not  as  good  as  those  previously  found.  Since  this 
value  of  y9  exceeds  yl9  y^  ought  to  exceed  15°.  Making 

y,  =  17°,  //,  =  0.10  =  /i,, 
and  other  quantities  as  before,  we  find 

Rev.  =  880;  E  =  0.8361 ;  yl  =  0.872,  y,  =  0.869, 

where  the  ratio  of  y1  -f-  ya  is  good,  but  the  other  values  are  too 
large. 

In  regard  to  the  value  of  or,  measurements  on  Francis  plates 
(Lowell  Hydraulic  Experiments]  show  that  a,  as  it  is  measured, 
will  not  exceed  22°,  and  yet  Rodmer  gives  the  mean  angle  a  as 
28°.  We  solve  the  problem  with  Rodmer's  values  y9  =  22°, 
a  =  28°,  assuming  /*,  =  0.10,  ^  =  0.05  ;  also  /*,  =  0.10  =  /;„ 
with  the  following  results  : 


78 


HYDRAULIC    MOTORS. 


TABLE  XI. 

rt  =  3.38  ft.,  r,  =  4.0  ft.,  yl  =  90°,  a  =  28°, 
y,  =  22°,  11  —  12.9  ft.,  Q  =  136  cu.  ft. 


Mi  =  0.10 
M2  =  0.05 

Mi   =  0.10 

M2  =  0.10 

Rev  per  second. 

0  869 

0  8636 

-^max  Per  cent  

81  96 

79  30 

«i  feet  per  second  

9  822 

9  755 

#2  feet  per  second 

23  525 

23  059 

V    

20  9012 

20  770 

Vi  Ea   (141)  .  . 

0  668 

0  673 

Vo  Eq.  (141).  . 

0  650 

0  663 

7/,/W0 

1  027 

1  015 

The  brake  efficiency  of  this  wrheel  being  0. 79375,  its  hydraulic 
efficiency  will  probably  be  81  or  82  per  cent.,  so  that  the  last 
hypothesis  above  is  not  admissible.  Rodmer's  values  for  a  and 
yt  with  low  resistances  give  a  satisfactory  result  for  the  effi- 
ciency, and  about  1  revolution  more  per  minute  than  the  meas- 
ured one  ;  but  they  do  not  produce  the  proper  values  for  the 
depths  yl  and  ?/2. 

The  total  least  section  of  the  buckets  is 

&'3  =  7.687  sq.  ft., 
and  if  the  velocity  be  v^  —  23.5,  the  depth  would  be 

136  7     f 

-  23.5  X  7.687  sq.  ft.  ~ 

so  that  the  proper  depth  cannot  be  found  with  this  data,  and  it 
would  require  a  coefficient  of  contraction  of  0.76  to  give  0.93  +. 

The  total  least  section  of  the  guides,  as  measured,  was 
K'  =  6.5371  sq.  ft.  ;  the  total  initial  arc  of  the  buckets  was 
k\  =  19.380  sq.  ft. 

To  show  the  effect  of  a  large  terminal  angle  of  the  bucket,  a 
computation  was  made  with  the  data 


HYDRAULIC    MOTORS.  79 

r,  =  3.38,  ^=4.00,  a  =  22°,  y,=  90V,  =  /<a  =  0.10,  and  yn_  =  28°, 
and  the  result  gave 

ErfMX.  =  0.738,  Rev.  =  0.850, 

an  efficiency  nearly  6  per  cent,  less  than  the  observed,  at  about 
the  same  speed.  "With  the  same  data,  except  j.il  =  0.20,  //2  — 
0.05,  it  was  found  : 

EM*.  =  0.780,  Rev.  =  0.820, 

so  that  with  less  revolutions  the  efficiency  approximated  to  the 
actual. 

As  seen,  the  method  of  "  trial  vane  angles"  may  be  made  to 
fit  the  conditions  of  the  wheel  when  the  results  are  known  ;  but 
as  a  method  of  design  it  is  not  satisfactory,  if  not  worthless. 
The  "  least  section"  as  used  by  Francis  is  better,  but  the  condi- 
tions of  that  cannot  be  realized,  for  the  stream  cannot  change 
suddenly  from  the  tangent  al>'  to  a  £.  AVe  now  suggest  the  fol- 
lowing, which,  though  not  perfect,  offers  the  most  promising 
condition  for  a  practical  solution.  Draw  the  tangent  a  //,  Fig. 
200,  and  through  it,,  the  middle  of  the  arc  a  d,  erect  a  perpendicu- 
lar^ r  to  a  T)'\  it  will  be  found  that  in  this  case  p  is  the  point  of 
tangency  to  p  d  of  a  line  parallel  to  a  I)'.  A  particle  at  a  would, 
if  free,  pass  off  in  the  direction  a  7/  tangent  to  the  bucket,  and 
would  trace  that  tangent  on  a  plane  rotating  with  the  crowns. 
Similarly,  a  particle  at  Ji  would,  if  free,  go  off  in  a  tangent  at  li ; 
but  this  tangent  is  not  parallel  to  a  V .  Similarly  at  all  points  in 
a  h  the  particles  would,  if  free,  move  in  tangential  directions  to 
their  paths  ;  but  those  tangents  all  have  varying  directions.  The 
particles,  however,  are  not  free.  Those  moving  along  the  back 
of  the  bucket  tend  continually  to  leave  it,  and,  with  other  particles 
in  a  h,  force  the  stream  toward  a  I',  and  the  stream  may  leave 
the  back  of  the  bucket  before  reaching  p,  while  the  mass  of  water 
in  the  vicinity  of  a  opposes  such  deflection,  the  result  being  a 
contraction  of  the  stream.  If  the  bucket  be  full  at  exit,  the 


80  HYDRAULIC    MOTORS. 

coefficient  of  contraction  may  be  found.  The  middle  line  of 
the  bucket  will  pass  through  u,  and  at  u  the  terminal  angle  y9 
will  be  the  same  as  at  a  (10°).  With  the  data, 

rt  =  3.375,  >-2  —  4.146  (which  call  4.1,  since  the  buckets  do 

not  extend  to  the  outer  rim)  ; 
a  =  21°,  Yl  =  90°,  y,  =  10°  ; 
/i,  =  0.10,  //2  =  0.20,  Q  =  136, 
we  recompute,  finding 
Rev.  per  sec.  0.867. 
-Z^max.  per  cent,  81.18. 
vl9  feet    7.072. 
tfa,  feet  22,57. 
y1?  feet    0.93. 
?/2  by  equation  (141)  would   exceed  1.3 ;  but  we  will  find  it 

as  follows  : 

Measure^*  r  on  the  drawing,  finding 
p  r  =  0.166, 

and  assume  that  the  value  of  vt  —  22.57  is  the  computed  mean 
velocity  of  the  section,  and  assume  (for  trial)  a  coefficient  of 
contraction  of  0.88  ;  then 

v,  X  0.88  (44  X  0.166)  y,  =  136 ; 
.  •  .  y2  —  0.9304  feet, 

=  0.93  to  the  nearest  hundredth, 

which  is  not.  only  sufficiently  exact,  but,  in  our  ignorance  of  the 
depth  of  the  stream,  may  be  actually  too  large  ;  hence  the  co- 
efficient 0.88  is  sufficiently  small.  If  the  coefficient  of  discharge 
be  0.90  the  computed  depth  will  be  0.9166  feet,  which  is  about 
•J-  of  an  inch  less  than  the  measured  depth  between  the  crowns, 
and  this  may  possibly  be  greater  than  the  depth  of  the  live 
stream  at  that  point. 


HYDRAULIC    MOTORS.  81 

The  exact  character  of  the  stream  in  the  buckets  is  unknown, 
especially  at  the  terminus,  so  that  perhaps  the  computed  depth 
ought  not  to  agree  exactly  with  that  of  the  wheel,  but  it  is  very 
certain  that  the  initial  end  of  the  bucket  was  full,  with  gate  fully 
open,  for  with  the  least  closing  of  the  gate  there  was  a  diminu- 
tion in  the  volume  discharged. 

Many  other  computations  have  been  made,  but  they  add  little 
if  anything  to  our  knowledge.  As  a  result  of  this  study  it  is 
inferred  that  for  an  outflow  turbine  similar  to  that  of  the  Francis, 
the  efliciency  may  be  determined  quite  accurately  by  equation 
(16#),  using  the  dimensions  of  the  wheel,  including  the  guide  and 
bucket  angles  ;  and  the  initial  depth  between  crowns  to  deliver  a 
given  amount  of  work  may  be  determined  by  the  second  of 
equations  (20) ;  but  the  terminal  depth  is  more  accurately  deter- 
mined by  means  of  a  measurement  across  the  stream,  as  shown 
above.  It  will  be  seen  hereafter  that  the  same  general  principles 
apply  to  other  turbines. 

No  motor  is  designed  which  secures  exactly  a  required  result. 
There  are  minor  elements  which  cannot  be  figured  out  exactly. 
After  all  the  labor  and  study  which  has  been  given  to  the  steam 
engine,  it  is  necessary  to  "  test  "  one  to  determine  exactly  what 
it  will  do,  and  it  is  the  same  with  other  motors.  The  perform- 
ance of  a  wrell-proportioned  turbine  may,  apparently,  be  deter- 
mined as  accurately  by  theory  as  that  of  any  other  motor. 

46.  In  this  wheel  the  inside  of  both  the  crowns  is  convex  in- 
ward, as  shown  in  the  figure,  the  curvature  being  such  as  to  re- 
duce the  depth  between  the  crowns  f  of  an  inch  at  5£  inches 
from  the  inner  rim.  This  reduction  is  about  y1^  of  the  initial 
depth.  No  reason  is  assigned  for  this  form.  If  the  object  of 
this  curvature  was  to  conform  to  the  form  of  the  vena  con- 
tracta,  it  appears  to  be  too  small.  In  any  case,  its  effect  will  be 
to  increase  the  velocity,  and  hence  reduce  the  pressure  in  the 
wheel  at  that  point ;  but  if  the  internal  pressure  exceeds  the  ex- 


82 


HYDRAULIC    MOTORS. 


ternal  pressure  (the  external  being  that  of  an  atmosphere  2116 
pounds  per  foot,  and  the  weight  of  2.2  head  of  water  in  the 
wheel  pit  about  138  pounds,  or  a  total  of  2254  pounds)  it  seems 
to  be  unnecessary.  The  internal  pressure  was  computed  from 
equation  (45),  page  22,  which  reduces  to 

P  =  Pa  +  #  h,  4 


)  sin.  Yi  +; 


sin2  (a  +  Yl) 
with  the  following  results  for  mid-section 

TABLE  XII. 


,)--  f]  sin 


(143) 


If 

Then 

Pressure  Ibs.  per 
Square  Foot. 

Pressure  Ibs.  per 
Square  Inch. 

r-* 

*i  =  1.205 

2,637 

18.31 

y  =  0.9  #! 

^  =  1.339 

2,630 

18.26 

y  -  0.8  y! 

T  =  L51° 

2,610 

18.12 

y  =  0.7  #1 

^  =  1.720 
A/* 

2,581 

17.92 

y  =  0.6  i/i 

^  =  2.01 
/c 

2,540 

17.64 

y  =  0.5.  y, 

^1  =  2.41 

2,380 

16.52 

In  all  cases  where  k  is  involved  it  is  assumed  that  the  stream 
completely  fills  the  successive  sections,  otherwise  k  ought  to  be 
the  actual  section  of  the  flowing  stream  : 

But  it  is  more  important,  in  this  case,  to  find  the  pressure  at 
other  points  in  the  wheel.  We  have  found  the  pressure,  p^  at 


the  entrance  into  the  bucket  where  p  = 


i\,  and  at 


the  width  of 


HYDRAULIC    MOTORS.  83 

the  crown  from  the  inner  rim,  at  -J,  f- ,  and  J,  giving  pounds  per 
square  foot  as  follows  : 

Pi  P\  P\  Pi  Pi  P* 

2648          2670          2667        2610          2420          2254 

ug  =  0.986,  ^  =  1.20,  -r-1  =  1.67,  p  =±  2.2,  ^  =  3.3. 

The  pressure  p^  is  at  the  gate,  where  the  velocity  is 

Y  =  7.02  4-  sin  21°  =  19.6  feet. 
At  J  the  width  of  the  crown  the  velocity  will  be 

v  =  7.02  ><  0.986  =.-  6.29  feet, 

and  the  pressure  is  greater  than  at  the  gate.  Since  the  terminal 
velocity  is  about  3.3  times  the  initial,  the  velocity  ought  to  in- 
crease continually  from  the  entrance  into  the  bucket,  to  the 
exit,  and  therefore  the  normal  sections  should  continually  de- 
crease, which,  as  is  seen,  may  not  be  the  case  with  flat  crowns 
and  partitions  of  uniform  thickness.  Thus,  as  shown  just  above, 
the, initial  section  \  is  0.986  of  the  section  at  one-fourth* the  width 
of  the  rim  from  the  initial  element ;  hence  the  normal  sections 
increase  at  first,  and  then  decrease.  To  secure  a  continual  dimi- 
nution of  sections,  the  depths  of  the  buckets  may  at  first  diminish 
and  later  increase  by  curving  the  crowns,  as  in  Fig.  21,  the  least 
depth  being  near  the  inner  rim  ;  or  still  more  accurately  as 
shown  in  Fig.  54«.  If  the  wheel  be  cast  the  diminution  of 
section  may  be  secured  by  increasing  the  thickness  of  the  wTalls, 
as  in  Fig.  22,  the  crowns  being  plane.  If  the  terminal  angle  be 
less  than  10  degrees,  the  outer  depth  (in  this  wheel)  should  be 
greater  than  the  inner,  and  the  crowns  should  flare  outward,  as 
in  Fig.  23.  The  wheel  may  be  so  submerged  as  to  produce  any 
desired  terminal  pressure.  If  the  Tremont  wheel  were  sub- 
merged 394  -^  64.2  =  6.1  feet  more  (p.  73),  or  8.3  feet  below  the 
surface  of  the  water  in  the  wheel  pit,  then  p^  —  p^  or  the  pres- 
sure at  the  ends  of  the  bucket  would  be  equal. 


84  HYDRAULIC    MOTORS. 

47.   The  direction  of  the  water  as  it  leaves  the  outer  rim  is 
given  by  equation  (35), 


cot  6  =  cot  y    --  —  . 


^  sin  yz 
where  B  is  the  angle  measured  from  the  rear  arc.     This  gives,  in 

this  case, 

8  =  91°  33', 

or  the  water  will  be  thrown  forward  of  the  radius  prolonged  1 
degree  33  minutes  ;  hence  the  direction  should  be  nearly  radial. 
Francis  found,  by  a  movable  vane,  about  34  degrees  ;  but  it  is 
difficult  to  account  for  so  large  an  angle,  and  one  is  inclined  to 
think  that  there  was  a  defect  in  the  measurement.  The  measured 
angles  appear  to  be  very  erratic.  It  is  asserted  by  some  writers 
that  the  quitting  direction  should  be  radial  for  best  effect,  but 
theory  and  experiment  unite  in  showing  that  the  assumption  is 
not  true,  except  in  special  cases. 

According  to  the  latter  part  of  Article  45  the  particles  of  a 
finite  stream  at  quitting  will  not  pass  off  in  parallel  lines,  and 
may  account,  at  least  in  part,  for  the  discrepancy  between  the 
computed  and  measured  angles. 

48.  The  radial  component  of  the  velocity  at  quitting  will  be 

v,  sin  y,  =  5.990, 

or,  say,  6  feet  per  second.  The  velocity  of  the  quitting  water 
will  be  F~2  in  equation  (34),  which  is 

F2  sin  B  —  v<>  sin  y9  ; 

.  •  .  F2  =  5.99, 

which  is  nearly  the  s'ame  as  the  radial  component,  as  it  ought  to 
be,  since  the  direction  is  nearly  radial  —  call  it  6  feet. 

The  initial  velocity 

v-  =  7.1, 

or,  say,  7  feet,  is  radial  in  this  case,  so  that  the  radial  component 
of  the  velocity  diminishes  from  about  7  feet  to  about  6  feet.  For 
intermediate  points  it  will  depend  upon  the  ring  sections. 


HYDKAULIC    MOTORS.  85 

49.  To  find  the  diameter  of  the  shaft. — The  shaft  will  be 
subjected  to  a  twisting  stress.  Let  P  be  a  twisting  force,  and  a 
its  arm  in  feet ;  n,  the  number  of  revolutions  of  the  wheel  per 
minute;  II. P.  the  number  of  horse-powers  delivered,  then 

P.  2  nan 


33,000 


=  JB.P. 


.. 

n  16 

where  d  is  in  inches,  J  the  modulus  of  rupture  to  torsion  = 
of  50,000,  say.  _ 


.  •  .  d  =         100  H'P'  nearly.  (144) 


If  H.P.  =  161,  n  =  52,  then  d  —  6f  inches,  nearly. 

In  the  Tremont  wheel,  the  diameter  of  the  shaft  was  7  inches 
from  the  wheel  to  the  upper  bearing,  and  larger  in  the  hub.  This 
gave  a  large  margin  for  safety,  the  factor  being  more  than  16. 

50.  To  find  the  path  of  the  water  in  reference  to  the  earth.— 
Divide  the  space  between  the  outer  and  inner  rims  into  any 
number  of  parts,  by  concentric  circles.  Suppose  there  are  8 
equal  parts? 

rt  =  4.2 

r,  =  3.38  /< 

8)0.82 


0.1025  feet  between  consecutive  rings. 
Let  />„  p2,  etc.,  be  the  radii  of  the  successive  annuli,  then 

P2  —  pl  =  p,  —  r,  =  0.1025  feet. 

The  initial  velocity  being  radial  and  v1  =  7.0  feet,  the  time 
required  to  go  from  the  inner  circle  to  the  next  will  be,  consider- 
ing the  velocity  uniform  over  this  space, 


86  HYDRAULIC    MOTORS. 

but  in  this  time  the  point  al  of  the  bucket.  Fig.  24,  will  have 
gone  forward  a  distance 

#,  d^  =  pl&>Tl  =  0.27  feet. 
The  radial  velocity  at  the  second  arc  will  be 

--^ -5- Bin  r,     JA  !>Lj/i_!;i        rf      (       } 

-  ^  5  sin  r       y  Ply 

in  which  ^  and  y  must  be  measured  from  the  elevation  in  Fig. 
19.  It  will  be  a  little  more  accurate  to  take  v,  midway  between 
rl  and  p,,  and  y  midway  between  p,  and  p2 ;  but  as  the  working  of 
the  wheel  and  the  efficiency  do  not  depend  upon  the  path  of  the 
water  in  reference  to  the  earth,  and  only  gives  some  information 
in  regard  to  the  course  of  the  water,  the  method  given  is  con- 
sidered sufficiently  accurate.  Then 

T    .  .   P2  "  Pi 

±  2       ? 

vr , 
and 

«,  <7,  =  A  «  (T,  +  T,). 

and  so  on  through  the  wheel.  It  will  be  seen  from  Fig.  24  that 
the  terminal  direction  is  nearly  radial. 

A  still  more  simple,  but  approximate,  method,  is  to  consider 
the  diminution  of  the  radial  velocity  as  uniform.  The  initial 
being  7  feet  and  the  terminal  6  feet  (say),  the  diminution  of 
velocity  from  one  annulus  to  the  succeeding  one  will  be  £  of 
(T  -  6)  =  0.125  feet,  Then 

^=0.1025-7-7;  Ta=0.1025+-6.875  ;  rs=0.1025-f-6.750,  etc. 
(The  Tremont  turbine   run  continuously  from   1849  to  1892, 
when  it  was  removed  to  give  place  for  another  wheel   of  larger 
power.) 

51.  The  Bucket. — In  the  study  of  the  Tremont  turbine,  some- 
thing was  said,  in  Article  45,  in  regard  to  the  form  of  the  bucket, 
but  it  is  such  an  important  element  that  more  may  properly  be  said. 
The  parallel  flow  wheel  offers  the  simplest  solution.  Let  the  hori- 


HYDRAULIC    MOTORS.  87 

zontal  lines  through  a  and  <?,  Fig.  25,  represent  respectively  the 
lower  and  upper  crowns  of  a  parallel  flow  wheel,  and  the  one 
through  g  the  upper  limit  of  the  guides.  The  number  of  buckets 
being  assumed  and  the  diameter  of  the  wheel  given,  the  distances 
,a  b  =  1)  c,  etc.,  will  be  found.  At  «,  &,  <?,  etc.,  lay  off  lines  a  d, 
•etc.,  making  angles  j\  =  15°  or  whatever  value  is  assumed. 
From  1)  erect  the  perpendicular  b  d,  and  prolong  it  to  e  /  with  e 
as  a  centre  and  radius  e  d  describe  the  arc  df,  and  similarly  with 
the  other  buckets.  This  will  make  yl  =  90°  If  it  is  desired 
to  make  yl  more  than  90°,  the  centre  of  the  arc  must  be  between 
e  and  d  •  and  if  less  than  90°,  the  centre  must  boon  ed  pro- 
longed. In  a  similar  manner  for  the  guides,  lay  off  lines  making 
angles  a  —  25°  or  whatever  value  is  assigned  ;  draw  the  perpen- 
dicular f  A,  prolong  to  </,  and  with  g  as  centre  and  radius  g  h  de- 
scribe an  arc,  and  similarly  for  others.  This  is  a  simple  mode  of 
constructing  the  lines  for  the  buckets  and  guides. 

If  yl  is  obtuse,  the  normal  section  of  the  bucket,  if  the  walls 
are  of  uniform  thickness,  will  increase  from  the  initial  end  to 
some  point  within  the  wheel  and  beyond  that  decrease.  This,  as 
has  been  stated,  is  an  objectionable  feature.  This  objection  may 
be  removed  by  making  the  back  of  the  vane  of  suitable  form,  as 
in  Fig.  26,  and  to  make  it  light  as  possible  it  may  be  "  cored." 
This  gives  the  appearance  of  a  double  vane,  and  is  called  a  "  back 
vane,"  a  feature  first  introduced  by  Haenel,  a  constructor  of  tur- 
bines, about  the  year  1848.  A  "  back  vane"  considered  by  it- 
self is  objectionable  on  account  of  its  weight  and  difficulty  in 
construction.  At  the  present  time  most  turbines  are  made  with 
an  initial  angle  of  the  bucket  of  90°  or  less,  for  which  there  is 
little  need  of  back  vanes,  and  they  are  now  rarely  made. 

The  surfaces  of  the  vanes  may  be  generated  by  a  radius  having 
as  line  directrices  the  axis  of  the  shaft  and  one  of  the  curves  in 
Fig.  26,  placed  at  the  circumference,  and  the  plane  of  the  base 
of  the  wheel  as  a  plane  directrix.  The  surfaces  will  be  warped 
— helicoidal. 


88  HYDRAULIC    MOTORS. 

When  so  generated  the  slope  of  the  surface  nearer  the  shaft 
will  be  greater  than  that  more  remote.  By  the  use  of  back  vanes 
this  may  be  avoided,  and  almost  any  desired  form  of  bucket 
made.  The  varying  slope  produces  a  difference  in  the  mechani- 
cal action,  and  centrifugal  force  causes  the  outer  part  of  the 
bucket  to  be  full  when  the  inner  part  may  not  be.  For  these 
and  other  reasons  the  buckets  are  sometimes  divided  into  two  or 
three  smaller  ones  by  annular  rings.  Fig.  27  is  the  plan  of  a 
bucket  of  uniform  width. 

Attempting  a  similar  process  for  the  radial  outflow  wheel,  we 
proceed  as  follows  : 

Let  a  b  =  b  c  =  c  d  =  width  of  bucket,  Fig.  28.  Through 
a,  5,  c  draw  lines,  making  angles  equal  to  yv  From  l>  erect 
perpendicular  b  e,  and  with  o  as  a  centre  on  ~b  e  prolonged  and 
o  e  as  radius  describe  the  arc  e  I'  I  i,  and  similarly  for  the  other 
buckets.  Then  from  I  draw  Ip  perpendicular  to  o  j  as  a  partial 
back  vane,  and  similarly  for  the  others.  The  width  of  such  a 
bucket  is  variable,  increasing  from  Ij  to  I'  m',  thence  decreasing 
to  f  and  again  increasing  to  the  end,  which  feature  is  objection- 
able. If  the  initial  angle,  y^  is  to  be  90°,  b  e  must  be  prolonged 
to  meet  *  0  tangent  at  i,  where  the  point  i  may  be  found  by  trial. 
The  buckets  may  be  made  of  uniform  width  as  follows :  Lay 
off  the  terminal  angles,  y^  at  a,  £,  e,  Fig.  29,  as  before.  At  0,  a 
small  arbitrary  distance  from  J,  erect  the  perpendicular  e  o,  and 
describe  e  f.  Similarly  with  o'  as  centre  describe  h  i.  From 
f  erect/"  I  perpendicular  to  o'  h,  and  with  o'  as  centre  describe 
p  I  to  the  intersection  with  o'  i,  and  from  p  draw  p  I  parallel 
to  c  i.  The  thicker  part  of  the  partition  wall  may  be  cored  as 
indicated  in  the  figure.  Since  the  Boyden  wheel  with  Russia 
sheet-iron  plates  for  buckets,  and  the  Swain  and  Hercules  tur- 
bines, hereafter  described,  with  cast-iron  buckets,  give  high 
efficiencies  and  are  durable,  it  is  hardly  necessary  to  give  further 
discussion  on  forms. 


Guides 


JSn  chcls 


FIG.  26. 


HYDRAULIC    MOTORS.  89 

To  find  the  depth  of  the  buckets,  when  of  uniform  breadth, 
we  have  in  all  cases  of  continuity, 

lev  —  k,  vl  =  &2  -ya, 
and  since  the  breadth,  i  p,  Fig.  29,  in  this  case  is  uniform, 

y  v  =  y,  V*  =  y,  ^>  (146) 

and  if  -y  increase  by  equal  increments,  y  will  decrease  correspond- 
ingly, and  the  equation  will  be  an  equilateral  hyperbola,  in  which 
the  axis  of  v  is  the  developed  arc  of  the  bucket.  As  shown  in 
Tables  VI.,  VII.,  IX.,  X.,  v2  exceeds  vt\  hence  the  terminal 
depth  should  be  less  than  the  initial  for  this  case.  This  form  of 
bucket  has  not,  so  far  as  known  to  the  writer,  been  used,  but  it 
seems  to  have  commendable  features. 

52.  The  Hoyden  d  iff  user  shown  in  Fig.  30  consists  of  a  coni- 
cal ly  diverging   stationary   piece,  A,   outside   the  buckets.     Its 
office  is  to  produce  a  diminishing  velocity  of  discharge,  and  thus 
impart  more  energy  of  the  water  to  the  wheel. 

53.  Turbine  at   Boott  Cotton  Mills,  Lowell,  Mass. — This  is 
an  inward  flow  or  vortex  wheel,  Fig.  31.     It  was  made  from  the 
design  of  Mr.  Francis  in  1849,  and   tested  by  him  in   1851,  and 
was  to  develop  230  horse-power  with  19  foot  head. 

The  regulating  gate  was  a  cylinder  of  cast  iron  placed  between 
the  guides  and  wheel,  and  was  made  water-tight  by  means  of 
leather  packing.  The  water  was  conducted  to  the  wheel  by  a 
wrought-iron  riveted  pipe,  about  130  feet  long,  8  feet  in  dia- 
meter, plates  f "  thick. 

DIMENSIONS  OF  THE  BOOTT  TURBINE. 

Outside  diameter,     .     .     .     ,.    ,     .     .  2/>,  =  9.338  ft. 

Inside         "  ;     .'    .     .     .     .     .  2/-,  =  7.987  " 

Least  depth  of  guide  passages,    ...  Y  —  0.999  " 

Outer     "       "  buckets,      .     .     .     .     .  yt    =  1.000  " 

Inside    "       "  buckets, y,    =  1.230  " 

Number  of  buckets N  =  40 

"         "     guides, 40 


90 


HYDRAULIC    MOTORS. 


Thickness  of  wheel  vanes,     ....     Jin.: 

"  "    guide      u          ....  TVn-: 

Terminal  angle  of  guides,  a     \ 

"  "      "    stream,  mean,  a'     . 

«  "      "    buckets,  .     .     .     .  ft 

Mean  angle  of  outflow  from  buckets,      y\ 
Initial     "      "    buckets,   .     t     ,     .     .     a 
Measured  area  of  guide  passages,  .     .     K' 

u  "     "    outflow  from  buckets,  k\ 

Calculated  least  area  of  buckets,      .     ,     &„ 
If  contraction  be  0.9  effective  area,     .     K" 

ii  it  ..(|<(  U  U  7.  " 


0.0208  ft 
0.0156  " 

8° 
12° 
10C 
15° 
62° 

5.904  sq.  ft. 
6.8092  "  " 
4.343 

5.3136  sq.  ft. 
6.1283  "  " 


Ratio  of  depths, 


&    =  0.82. 

y« 


TABLE  XIII. 
ABSTRACT  OF  EXPERIMENTS  ON  THE  BOOTT  CENTRE-VENT  WATER- WHEEL. 


| 

3 

' 

g 

1* 

3 

a 

1 

111 

1^ 

u 

1| 

Sco> 

£ 

h 

£ 

*    «   « 

2o»S 

g  J 

C  G 

a 

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C-^  ir,^, 

^^3 

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£ 

'QtH  C 

p. 

A 

O  02 

*3 

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*o  "^  fe 

r™* 

^         ife 

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<M    ^ 

H 
W 

I6 

2 

3 

11*5 

I8 

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§  fc    'o^ 

U 

03    , 

1 

53 

6 

o 

&2o 

£ 

S 

1 

"3  ° 

53 

5 

3 

30.0 

14.197 

67.029 

59351.7 

226:30.5 

0.38129 

14.647 

0.6847C 

6 

3 

27.0 

14.143 

66.889 

59002.2 

•32026.8 

0.37332 

13.185 

0.43714 

14 

6 

41.7 

13.778 

90.166 

77481.4 

45135.3 

0.58254 

20.372 

0.6  43? 

15 

6 

35.9 

13.606 

91.697 

77812.1 

46402.8 

0.59634 

17.540 

0.59291 

23 

9 

434 

13.332 

103.769 

86229.3 

620654 

0.71986 

20.861 

0.7123!; 

24 

9 

42.7 

13.334 

103.689 

86229.3 

62752.2 

0.72774 

20.497 

0.7006< 

25 

9 

41.9 

13.304 

10-1.229 

86483.7 

63199.3 

0.73007 

36.785 

1.2.390; 

27 

12 

42.6 

73.400 

112.525 

94057.5 

74979.3 

0.79716 

20.806 

0.7086? 

28 

12 

42.0 

13.431 

112.987 

94662.2 

75347.2 

0.79596     i     20.515 

0.6979( 

29 

12 

40.7 

13.331 

112.562 

93603.9 

74580.9 

0.79767    !     19.929 

0.6805£ 

30 

12 

40.3 

13.387 

112.996 

942964 

75153.2 

0.79699         19.711 

0.67135 

31 

12 

39.6 

13.386 

113.071 

94415.2 

75208.3 

0.79657         19.364 

0.6599C 

32 

12 

38.9 

13.383 

113.164 

94471.1 

75249.2 

0.79653         19.029 

0.(;485( 

33 

12 

38.2 

13.356 

113090 

94219.0 

75142 

0.79753         18.688 

0.6369 

34 

12 

37.4 

13.381 

113.673 

94881.9 

75103 

0.79754 

18.316 

0.62431 

35 

12 

36.8 

13.405 

114.293 

95571.2 

75202 

0.78132 

17.998 

0.61291 

FIG.  29. 


FIG   30 


HYDRAULIC)    MOTORS. 


91 


54.  Computations. — An  examination  of  Fig.  31  shows  that  a 
tangent  to  a  guide  at  its  terminus  falls  entirely  outside  the 
wheel  ;  hence  the  angle  or,  which  should  be  used  in  the  formula, 
is  indeterminate.  If  the  water  were  not  confined  none  would 
enter  the  wheel,  but  as  it  is  confined  and  the  supply  continuous, 
it  will  be  forced  into  the  wheel,  but  at  unknown  angles  some  fila- 
ments may  enter  at  90°.  It  is  really  a  case  of  discontinuity  as 
regards  our  formulas,  and  hence  they  do  not  strictly  apply  ;  know- 
ing, as  we  do,  the  least  section  between  the  guides,  5.904  square 
feet,  and  the  volume  discharged,  113  cu.  ft.  per  second,  the  velo- 
city of  exit  from  the  guides  will  be 

F=  113  -7-  5.904  =  19.14  feet  per  sec. 
The  velocity  of  the  wheel  for  best  effect,  experiment  33,  was 

GO'  r,  =  18.688  feet, 

and  since  y1  =  62°,  equation  (17)  will  give  a  •—  16°  about,  which 
is  somewhat  larger  than  the  mean  angle  given  above.     Then 

Fsin  a        19.14  X  0.27         ,  f 

v,  =  =  — —  o  leet  nearly. 

sin  y^  0.866 

But  this  is  an  inverse  process,  and  we  now  try  a  direct  process, 
assigning  to  /*,  and  yw2  large  values  on  account  of  the  resistances 
resulting  from  imperfect  conditions. 

TABLE  XI V. 

DATA. 

r,  =  4.67,  t,  =  4.00,  y,  =  15°, 

H=  13.356  feet,  .Q  =  113.09  cu.  feet. 


REQUIRED. 

a  =  15° 
Mi  =  0.20 
M2  =  0  20 
Vl  =  60" 

a  =  12° 
Ml=0.25 
Ms  =  0.20 
yj  =  62* 

Measured     • 
Values. 

Angular  velocity  Eq  (16)  GO 

4.442 

4  288 

3.996 

Rev  per  minute  

42.42 

4095 

38.16 

Efficiency,  Eq.  (16^),  E  per  cent  

82.50 

81.06 

79.753 

Initial  velocity  (18)  v\ 

5.557 

4  33 

Terminal  velocity  (19)  v9     

17.339 

17  03 

Inner  depth  (141)  y\  

0.828 

1.062 

1.000 

Outer  depth  (141),  y2  

1.150 

1.181 

1.230 

Va/Vi  •  - 

1.37 

1.112 

1.230 

92 


HYDRAULIC    MOTORS. 


A  "  BOYDEN  "  TURBINE. 

55.  In  1882  a  Boy  den  turbine  of  the  Fourneyron  type  was 
tested  for  the  Merrick  Thread  Company.  This  was  several  years 
after  the  construction  of  the  most  celebrated  Boyden  wheel. 
The  following  are  the  dimensions  of  the  wheel,  and  results  of 
the  test : 

Internal  diameter,  . 2/\  =  73.60  in. 


External  v    .     .     .     .     ...  2>-s 

Terminal  vane  angle,       .     ....  a 

Mean  exit  angle,    .     ...     .     .     .  a' 

Initial  angle  of  bucket,   .     .     .     ,     .  yl 

Terminal  angle  of  bucket,  .     .     .     .  y2 

Mean  angle  of  flow,    .     .     .     .     ,  .  .  ;/2 

Initial  depth  of  wheel,    .     .     ...  y1 

Terminal  depth  of  wheel,    .     .     .     .  y2 

dumber  of  buckets, J\T  —  34. 

"  guides, 54. 

Terminal  least  section  of  guides    .     .  K'  —  6.814  sq.  ft. 

"  "  "  buckets  .     .  k\  =  5.660  "     " 

TABLE  XV. 

ABSTRACT  OF  A  TABLE  OF  FRANCIS  EXPERIMENTS  WITH  A  BOYDEN 
WHEEL  (OF  FOURNEYRON  TYPE). 


90.00  " 

24° 

29° 

90° 

26° 

29° 

8.64  in. 

9.125  in. 


z 

> 

•^ 

t-  --^ 

orf 

fc; 

"c  g 

fe 

2  ^ 

I'gg^. 

g     ^ 

»2 

Is 

.^ 

^-g^ 

?"i  oS 

j|N 

•  f    ^ 

h 

ss 

g 

^'  S 

£  S1^ 

"S3 

tB<r*  '  ^ 

^                      £~ 

H                  >2 

KSK 

s 

$ 

1.000 

57.00 

16'.  61 

145.35 

216.79 

79.17 

0.560 

1.000 

59.33 

16.57 

146.18 

216.55 

78.82 

0.584 

1.000 

63.50 

16.60 

147.10 

222.04 

80.17 

0.624 

..000 

66.50 

16.62 

148.32 

220.28 

78.79 

0.653 

0.833 

55.67 

16.66 

142.04 

203.19 

73.71 

0.546 

58.75 

16.63 

142.82 

207.23 

76.93 

0.577 

0.773 

56.00 

16.74 

136.12 

188.94 

73.11 

0.548 

0.662 

55.75 

16.80 

128.86 

169.28 

72.42 

0.545 

0.442 

50.63 

17.10 

105.65 

113.36 

55.32 

0.461 

FIG.  31. 


HYDRAULIC    MOTORS.  93 

The  hydraulic  efficiency  would  probably  be  about  82  per  <?ent. 
Judging  from  the  analysis  of  the  Francis  wheel  and  the  com- 
paratively large  terminal  vane  angles  in  this  wheel,  the  prejudicial 
resistances  were  low  ;  probably  //I  =  /*2  =  0.10  about. 

56.  It  will  be  seen  from  the  preceding  tests  that  the  efficiency 
falls  off  quite  rapidly  as  the  gate  is  closed  more  and  more.     This 
is  a  peculiarity  of  the  pressure  turbine,  and  for  this  reason  is  ob- 
jectionable where  the  supply  of  water  is  not  sufficient  to  fill  the 
buckets  with  open  gate,  for  the  lack  of  such  a  supply  necessi- 
tates a  partially  closed  gate  to  secure  a  head  in  the  supply  cham- 
ber.    This  condition  of  things  induced  Fourneyron  to  divide  the 
wheel  into  several  chambers,  as  if  separate  wheels  were  placed  one 
above  the  other,  as  in  Fig.  32.    With  this  arrangement,  with  the 
gate  one  third  opened,  the  lower  third  would  work  as  if  it  were 
the  only  wheel ;  similarly  the  second  third,  and  finally,  when  the 
gate  was  completely  open,  the  three  parts  worked  as  one  wheel. 

57.  Fourneyron  Turbine  at  St.  Blaise. — One  of  the  first  tur- 
bines constructed  by  Fourneyron  was  erected  at  St.  Blaise,  and 
worked  under  the  high  head  of  354  feet,  developing  30  effective 
horse-power  at  2300  revolutions  per  minute. 

Outer  diameter  r^  =  12.99  inches. 
Inner         "         r,  =     7.47      « 

Later  two  larger  wheels  were  substituted  for  this  one  working 
under  the  same  head,  developing  60  horse-power  with  the  same 
number  of  revolutions.  The  outer  diameter  was  r2  =  21.66 
inches.  At  these  high  speeds  the  bearings  needed  renewing 
every  10  to  14  days.  They  discharged  above  the  water,  as  shown 
in  Fig.  33.  Later  these  were  replaced  by  tangent  wheels,  and 
still  later  by  Girard  turbines. 

58.  Parallel  flow  Turbines. — The  parallel  flow  wheels  are 
usually  classed  as  Jonval  or  Girard,  which  are  constructed  sub- 


HYDRAULIC    MOTORS. 

stantially  alike  ;  but  the  former  is  buried  or  submerged,  while  the 
latter  discharges  above  the  water  in  the  wheel  pit,  Fig.  34.  The 
French  engineers  Gallon  and  Girard,  about  1856,  began  to  design 
impulse  turbines  for  all  possible  conditions  —  high  and  low  falls, 
large  and  small  quantities  of  water,  axial  and  radial  flow,  with 
horizontal,  vertical,  and  inclined  axes.  All  turbines  in  which  the 
velocity  from  the  supply  chamber  into  the  wheel  was  that  due  to 
the  head,  which  we  have  called  wheels  of  free  deviation,  were 
called  impulse  turbines,  and  in  Europe  every  variety  of  impulse 
turbine  frequently  goes  by  the  name  of  "  Girard."  He  was  the 
first  to  ventilate  the  buckets,  so  that  the  pressure  in  the  wheel 
would  be  that  of  the  atmosphere,  Fig.  35. 

A  A,  Fig.  34,  are  buckets  placed  between  parallel  crowns  and 
supported  by  radial  arms  or  segments  attached  to  the  shaft  ;  jE>  B 
are  guide  vanes,  directrices,  or  distributors.  The  cross-section  of 
Girard's  turbines  are  bell-mouthed  ;  but  as  first  made  by  Jonval, 
the  sides  were  parallel,  as  shown  in  Fig.  36.  The  Jonval  turbine 
may  be  placed  at  any  point  between  the  level  of  the  water  up 
stream  and  that  in  the  tail  race,  provided  there  be  a  closed  tube, 
called  a  "  suction  tube,"  through  which  the  water  passes  before 
being  discharged,  as  in  Fig.  36. 

59.  The  "  Collins"  Turbine.—  The  Collins  turbine  is  the  only 
parallel  flow  wheel  of  American  make  for  which  we  have  the 
results  of  a  test.  It  was  tested  in  1SS3. 

DIMENSIONS. 


«  =  iTi°,  r 

Mean  diameter,      ...     ,  '".  '  S     .  2>'m  =  4.170  ft. 

Terminal  depth  of  guides,    .....    ..  Y   =0.836    " 

Number  of  buckets,    ......  N   =  24 

"         "    guides,     ......  N,  =  30 

Measured  terminal  arc  of  guides,      .  K'  =  2.912  sq.  ft. 

"  "         "     "    buckets,    .  k\    =  2.882  " 


FIG.  32, 


FIG.  33. 


HYDRAULIC    MOTORS. 


95 


TABLE  XVI. 

ABSTRACT  OF  TEST  OF   A  60-iNCH  COLLINS   TURBINE  SUBMERGED  IN  THE 
WHEEL  PIT  (JONVAL  TYPE). 


, 

*fe 

»* 

.< 

§5 

fl 

"rt 

a;  O/O 

jf*  —     • 

^  C 

Ofl 

•          |    |1^* 

ft  0 

0  fc 
w  M 

E- 

3 

HI* 

lip 

|5  t-  ^ 

1!" 

P  S 

O 

EH 

B2 

K  A 

1.000 

16.57 

64.35 

96.41 

79.85 

85.25 

0.573 

1.000 

16.55 

64.40 

97.03 

80.40 

89.17 

0.600 

1.000 

16.56 

64.40 

96.65 

80.03 

93.00 

0.625 

0.748 

.79 

58.50 

86.69 

77.49 

81.50 

0.544 

0.600 

.86 

53.61 

69.53 

67.93 

77.90 

0.650 

0.503 

17.18 

45.14 

56.22 

64.02 

69.67 

0.460 

0.303 

17.41 

34.25 

34.72 

51.42 

71.00 

0.218 

0.161 

17.84 

24.80 

19.12 

38.16 

53.00 

0.984 

The   same   turbine   was  tested  with  a  suction  tube,  with  the 
following  results : 


TABLE  XVII. 
SIXTY-INCH  COLLINS  TURBINE  WITH  A  SUCTION  TUBE. 


i 

- 

IJl. 

fc 

ft 

.2   £ 

I   ^5 

H  « 

^^ 

1  j3  ^^ 

«  •^ 

*5  "^ 

a* 

l« 

^ 

§ 

j|!  r 

|S 

1* 

V 

O 

Mi 

P5 

1.000 

16.56 

64.88 

102.18 

84.01 

0.604 

1.000 

16.55 

64.99 

102.70 

84.34 

0.646 

1.000 

16.56 

64.88 

101.25 

83.25 

0.709 

0.548 

17.01 

68.87 

64.83 

65.97 

0.457 

The  same  turbine  was  also  tested  with  a  suction  tube,  in  which 
was  a  cone  with  its  base  placed  just  under  the  centre  of  the 
wheel,  so  as  to  act  as  a  diffuser,  with  the  following  results  : 


96 


HYDRAULIC    MOTORS. 


TABLE  XVIII. 
SIXTY-INCH  COLLINS  TURBINE  WITH  SUCTION  TUBE  AND  INVERTED  CONE. 


fe 

. 

gf 

i 

. 

1  1!^ 

o  6 

1  8,  • 

°^ 

K  2 

•< 

f 

•3- 

f 

ll 

w3  vw 

CJ 

PH 

tf 

n 

W 

1.000 

16.56 

65.18 

101.46 

83.05 

0.578 

1.000 

16.57 

65.30 

102.62 

83.76 

0.606 

1.000 

16.58 

65.37 

101.73 

82.92 

0.712 

0.548 

16.96 

52.09 

69.66 

69.66 

0.492 

The  efficiency  was  increased  with  the  suction  tube,  but  not  with 
inverted  cone. 

60.  A  96-inch  "  Collins  "  (parallel  now)  turbine  was  tested  at 
the  Holyoke  testing  station  about  1883,  which  gave  the  highest 
efficiency  of  any  recorded  experiments  with  a  parallel  flow  wheel 
so  far  as  we  know. 

NINETY-SIX  INCH  COLLINS  TURBINE  (JONVAL  TYPE). 
Gate  opening,    .     .....     .     .  fully  open. 

Total  head,  .     .     .     -.     ,    -.     .     •  '   .         H  —  16.59  ft. 
Volume  of  water  per  second,   ...         Q  =  113.46  cu.  ft. 
Number  of  revolutions  per  minute,  .         ^=63.38 
Brake  horse-power,    .     .....     .    HP  =  131.49 

Efficiency  of  wheel,  per  cent.,       .     .         E  =  85. 06 
Hydraulic  efficiency ,  possibly,      .     .     87  per  cent. 

61.  Segmented  Feed.- — A  turbine  works  with  better  effect  when 
the  buckets  are  full ;  and  when  the  wheel  is  too  large  to  secure, 
at  all  times,  full  buckets  with  variable  supply,  means  have  been 
devised  to  shut  off  a  part  of  them,  leaving  others  fully  open.    Fig. 
37  shows  such  a  device,  where  the  upper  part  of  the  guide  forms 
the  gate  by  sliding  downward  to  close  the  passage  of  the  guides, 
or  distributors.     Those  at  the  right  marked  B  are  fully  open, 
while  those  marked  A  at  the  left  are  shown  partly  closed.     They 
may  be  so  constructed  that  two  or  three  gates  may  be  closed  at  a, 


FIG.  34. 


FIG.  35. 


FIG.  36. 


HYDRAULIC    MOTORS.  97 

time,  while  the  others  remain  open,  and  so  be  adapted  to  great 
range  of  the  supply  of  water.  Segmental  feed  does  not  produce 
quite  as  efficient  a  wheel  as  full  feed,  but,  as  will  be  seen,  the 
difference  is  not  as  great  as  might  be  anticipated,  considering 
that  a  portion  of  the  water  will  have  but  little  effect  during  the 
tilling  and  emptying  of  the  buckets. 

62.  HaenePs  Turbine. — Haenel,  manager  of  machine  works  at 
Magdeburg,  constructed  several  parallel  floAv  wheels,  and  made  an 
extensive  set  of  experiments  which  are  very  instructive.  Ad- 
mission to  the  guide  wheel  was  regulated  by  a  pair  of  rubber 
strips,  supported  by  iron  stays,  and  rolled  upon  two  conical  roll- 
ers, the  object  being  to  admit  water  to  as  few  passages  as  desired, 
or  to  open  all  32  of  them  at  the  same  time,  Fig.  38.  A  A  are  the 
buckets,  B  B  the  guides.  C  C  conical  rollers,  E  E  the  rubber  flap. 
The  wheel  passages  were  designed  to  be  .of  equal  sections  normal 
to  the  flow  of  the  water,  so  that  the  velocity  through  the  wheel 
would  be  uniform  when  submerged.  -  This  was  accomplished,  in 
part,  by  the  use  of  back  vanes.  The  ventilating  pipes  connected 
with  the  vane  passages  were  sometimes  open  and  sometimes  closed, 
but  there  was  no  appreciable  difference  in  the  efficiency  due  to 
the  difference  of  the  two  conditions. 

The  turbine  tested  was  one  of  eight,  all  alike,  having  the  fol- 
lowing dimensions.  The  dimensions  in  feet  are  approximate,  as 
the  original  were  in  Prussian  feet,  and  for  our  purpose  it  is  not 
considered  necessary  to  be  particular  about  the  fractions  of  an 
inch  in  English  units  : 

1  Prussian  foot  =  1.02972  English  feet. 

1  square  Prussian  foot  =  1.06032  English  square  feet. 

1  cubic  Prussian  foot    =  1.09183  English  cubic  feet. 

DIMENSIONS  OF  HAENEL  's  TURBINE. 

Outer  diameter  of  wheel  and  guide  at  inflow,  5  ft.  9J  in. 

Inner  "  "  "  «  4  "  6     « 

Mean  "  «  "  "  d  =  5  "  1     " 


98 


HYDRAULIC    MOTORS. 


Outer  diameter  of  wheel  at  outflow,      .     .     . 
Inner  "  "  "  .... 

Initial  depth  (width)  of  wheel  buckets,      .     .     yl  = 
Terminal  "  "  .     .     ya  = 

Depth  of  wheel, /  «• 

Angle  of  outflow  from  guides, a  = 

Initial  angle  of  buckets, yl  = 

Terminal  angle  of  buckets  on  concave  side,    .     y^  = 

u  a  a  u    convex        « 

Measured  area  between  guides,     .     ....     .   K  = 

Effective  area  0.9  of  measured,    .     .    ...     .     .  K"  = 

"         "     of  buckets  at  outflow,      .     .     .  &a"  = 

TABLE   XIX. 
RESULTS  OF  TESTS  OF  HAENEL'S  TURBINE. 


6  ft.    5 

in. 

3  "   10 

(< 

1  "     3< 

L    a 

2  "      7' 

it, 

1  "       \ 

L    a 

22°  30' 

45° 

26°  20' 

23°  00' 

3.17  sq. 

ft. 

3.12  " 

u 

6.38  " 

a 

co 

h  3 

- 

1     .d 

_-J 

a 

u 

o  <5 

O     • 

c 

^  ^  ^ 

V^       • 

>    32 

C  "~ 

o 

££ 

K  Z 

£*  "32 

s 

,^  t£  "  ;^ 

v-i  **"  SH 

C  ^ 

^  H 

HH 

^ir  ^ 

~  "S 

c  3  s  - 

C  „  ^ 

^2nK 

S-S  g 

JE""! 

C  a 

NUMI 

GATE  01 

P 

1 

lit! 

*o      "~^ 

ll 

|^ 

2  » 

S| 

II 

| 

pq 

4 

6.65 

0.00 

5.80 

2250.8 

1457.7 

29.0 

0.6476 

0.7101 

8 

6.58 

0.11 

12.98 

5052.5 

3340.6 

27.5 

0.6612 

0.6876 

12 

6.48 

0.21 

19.80 

7540.9 

5351.2 

36.5 

0.7096 

0.1331 

16 

6.44 

0.21 

26.10 

10217.0 

6369.9 

35.0 

0.6816 

0.6982 

24 

6.18 

0.39 

47.61 

15844.0 

10556.0 

31.5 

0.6662 

0.6759 

32 

6.26 

0.29 

53.66 

19765.0 

1214.8 

72.5 

0.0615 

0.0703 

32 

5.90 

0.54 

62.20 

21579.0 

14703.0 

39.0 

0.6813 

0.6901 

Another  test. 

32 

4.07 

2.03 

48.93 

12060. 

8042.5 

32.0 

0.6669 

Still  another  tf  st, 

32 

5.12 

1.48 

45.66 

14156 

9676.2 

33.0 

0.6836 

0.6949 

FIG.  37. 


FIG.  38. 


HYDRAULIC    MOTORS. 


99 


According  to  these  results,  the  prejudicial  resistances  were  6J 
per  cent,  with  4  buckets  working,  and  was  about  1  per  cent,  with 
all  the  gates  open.  One  test  gave  only  j  of  one  per  cent,  prej- 
udicial resistance. 

63.  Tangential  wheels  are  radial  with  segmental  feed.     They 
are  more  especially  adapted  to  high  heads  with  a  limited  supply 
of  water.     They   are   made   larger   in    diameter   than  pressure 
wheels  of  the  same  capacity,  and  hence,  when  the  speed  of  the 
periphery  is  the  same,  the  revolutions  will  be  less  in  the  same 
time.    They  may  be  inflow,  as  in  Fig.  39,  or  outflow,  as  in  Fig.  40. 

64.  Industries  gives  an  interesting  account  of  the  turbines 
used  in  the  steel-manufacturing  establishment  at  Terni,   Italy. 
They  are  segmental  feed,  radially  outward  flow,  of  free  deviation, 
Fig.  40. 

TABLE  XX. 
SOME  DIMENSIONS  OF  TURBINES  AT  TERNI. 


HEAD. 
H. 

Horse-power. 
H.P. 

Volume  of  water. 
Q- 

Rev.  per  minute. 
n. 

Inner  diameter. 

»V 

Ft. 

Cu.  ft.  per  sec. 

Ft.        In. 

595.5 

1000 

19.77 

210 

7     10.5 

595.5 

800 

15.89 

200 

8      2.4 

595.5 

500 

9.89                  240 

6      5.9 

595.5 

330 

7.06 

200 

7     10.5 

595.5 

150 

3.00 

250 

6      4.7 

595.5 

50 

0.99 

850 

1     10.25 

595.5 

50 

0.99 

850 

1     10.25 

595.5 

40 

0.85 

450 

3      6.1 

595.5 

40 

0.85 

450 

3      6.1 

595  5                       30 

0.60 

600 

2      7.5 

595.5 

20 

0.42 

450 

3      6.1 

. 

65.  Dimensions  of  a  segmental  feed,  tangential  turbine, 
radially  outward  flow  of  free  deviation,  with  horizontal  axis, 
Fig.  40. 


100 


HYDRAULIC    MOTORS. 


2/'2  =  8  "  11.1  " 
T  =    4.32  in. 


y>    =  15.75  " 
N  =  110 
t      —  0.2  in. 

&  ft. 


Inner  diameter,      .    ' 2/1,  =  7  ft.  10  J  in. 

Outer         u  

Width  of  guide  passages,    .     . 

"         buckets,  initial,  . 

"  "       terminal,    . 

Number  of  wheel  vanes,     .     . 
Thickness  of  steel  vanes, 
Length  of  supply  pipe,   .     .     . 

(3481  feet  of  cast  iron,  remainder  wrought  iron.) 
Thickness  of  cast-iron  pipe,      .     .  0.71  in. 

Diameter       "  .     .  21.66  " 

"         of  wrought-iron  pipe, .  18.9     " 

Thickness  "  ...  0.20  "to 0.47 in. 

Designed  for, 400  H.P. 

With  head  of, H—  570.8  ft. 

And  water  supply  of,     .     .     .     .     Q  —  8.5  cu.  ft.  per  sec. 


TABLE  XXI. 

TEST  OP  THE  PRECEDING  WHEEL  AT  IMMEKSTADT.    (Arr  THE  TIME  OP  THE 
TEST  8.5  Cu.  FT.  WERE  NOT  AVAILABLE.) 


•I 

1 

h 

IV 

ti 

, 

BJj 

||  < 

f 

1  3  0°* 

l!i 

1 

Is 

0   $ 

ii 

6 

1 

O  s-, 

£ 

2 

«w 

10 

211 

570.84 

2.111 

136.8 

81.5 

0.595 

15 

211 

570.84 

3.350 

217.1 

144.3 

0.665 

20 

214 

570.84 

4.375 

283.4 

196.8 

0.694 

25 

216 

570.84 

5  187 

336.0 

253.9 

0.755 

31 

210 

570.84 

6.840 

440.9 

336.8 

0.764 

The  power  absorbed  by  the  friction  of  the  shaft  was  5  H.P. 
with  10-inch  opening  of  gate  and  4.5  with  25-inch  opening, 
or  about  3  per  cent,  of  the  total  power  in  the  former  case  and 


FIG  39. 


FIG.  40. 


HYDRAULIC    MOTORS.  101 

1  per  cent,  in  the  latter;  or  about  5J  per  cent,  of  the  brake  power 
to  less  than  1^  per  cent. 

Turbines  acting  under  a  fall  of  about  650  feet  are  described 
by  Knoke.  They  are  radially  outflow  and  of  cast  iron.  They  re- 
ceive water  through  two  guides  at  opposite  extremities  of  the 
diameter  (four  guide  passages  in  all). 

DIMENSIONS  OF  Two  OF  THEM. 

Inner  diameter,     .....     .     .  11.81  in.  11.81  in. 

Outer        «  .     ,     .     .     .     .     .  16.41  «  14.84  " 

No.  of  vanes,   ........  45  cast  iron  84  wrt.  iron. 

No.  rev.,      .     .     . 583.  928 

Yol.  water  per  min.,      .    ...    .     .     .  4.52  cu.  ft.  5.085  cu.  ft. 

Head,  feet, 129.  131 

Power  of  the  water,  d  Q  II,  ft.  Ibs.,  36440.  53806 

Brake  power,   ........  16498.  28809 

Efficiency,    ......;..  45.27.  53.5 

MIXED  FLOW. 

66.  Risdon  Wheel. — Mixed  flow  wheels,  so  far  as  known,  are 
inward  and  downward,  of  which  the  "  Risdon,"  Fig.  41,  is  a 
type,  giving  also  a  view  of  the  cross-section  of  the  wheel  jusfc 
under  the  upper  crown,  in  which  B  D  are  guides  and  A  buckets. 
The  illustration  shows  the  construction  so  clearly  that  a  detailed 
description  seems  unnecessary.  This  gave  the  highest  efficiency 
of  any  wheel  tested  at  the  Centennial  Exposition,  1876,  it  being 
reported  as  87  per  cent.  Those  tests  lasted  but  a  few  minutes, 
and  may  have  been  fortunate  in  showing  a  higher  efficiency  than 
they  would  maintain  for  a  long  period ;  but  this  wheel  has  main- 
tained a  reputation  for  high  efficiency,  and  in  some  more  recent 
tests  has  been  reported  as  giving  even  90  per  cent.  But  all  such 
high  figures  should  be  received  with  caution,  and  writers  dis- 
card the  last  figure  as  being  too  improbable  to  be  true. 


102  HYDRAULIC    MOTORS. 

67.  The  Humphrey  Wheel. — This  is  an  inward  and  downward 
flow  wheel,  Fig.  42.  The  wheel  A  A  has  13  buckets,  a  a  extend- 
ing below  the  casing.  C  C  is  a  regulator,  also  containing  12 
guides,  the  whole  moving  on  spherical  rollers,  and  operated  by  a 
hand  wheel,  G,  through  shafts  and  bevel  gearing  F.  By  this 
means  water  is  admitted  to  or  cut  off  from  the  wheel.  The 
wheel  is  enclosed  in  metal  having  curved  surfaces,  into  which  the 
water  was  conducted  by  a  riveted  pipe.  The  lower  end  of  the 
shaft  had  a  pivoted  step,  while  the  upper  end  was  supported  by 
collar  bearings.  Fig.  43  shows  a  plan  of  the  wheel  casing,  con- 
duit, and  brake  used  in  the  test.  The  diameter  of  the  brake 
pulley  was  5.44  feet;  width,  2.7  feet;  arm  of  brake  lever,  15.9 
feet.  The  wheel  was  a  design  of  the  "  Humphrey"  Machine 
Co.,  of  Xew  Hampshire,  and  was  designed  to  deliver  270  horse- 
power with  a  fall  of  13  feet.  It  was  tested  by  Mr.  James  B. 
Francis  about  1880.  The  head  was  determined  by  the  difference 
in  heights  of  the  water  in  two  tubes,  one  entering  the  supply  shaft 
above  the  wheel  and  the  other  in  the  tail  race  ;  and  the  quantity 
of  water  by  means  of  weirs  and  hook  gauges.  There  being  no 
contraction  at  the  ends  of  the  weir,  the  quantity  was  .computed 
from  the  formula 

Q  =  3.33  Z  7/f, 
where 

L  =  length  of  weir  in  feet  =  11.92  for  one  weir,  and  10.98 
for  the  other. 

H  =  head  of  water  above  sill  in  feet,  measured  some  dis- 
tance above  weir. 

DIMENSIONS. 

Outer  diameter,  2/'2  =  8.1  feet;  inner,  i\  =  2.0;  a  =  15°, 
yi  =  80°,  Y»  not  given,  but  the  result  of  the  test  indicates 
that  it  was  small,  probably  about  12°,  Y  =  2  feet. 


Fia  41. 


UNIVERSITY 


HYDRAULIC    MOTORS. 


103 


TABLE  XXII. 
TEST  OF  HUMPHREY  TURBINE. 

Brief  extract  of  results. 


OPENING  OF 
REGULATOR. 

W 

Volume  Water 
per  second. 

late 

?o^ 

|H,«o 

Brake  Power, 
U. 

Efficiency. 
E. 

fff'fe 

r«  ls 
3 

Per  cent. 
101.00 
88.50 
68.21 
52.90 
40.66 
27.94 
19.15 

ft. 
12.478 
12.839 
13.103 
13.456 
13.688 
14.069 
13.998 

cu.  ft. 
207.82 
181.59 
166.80 
131.52 
110.70 
79.26 
59.02 

Ft.  Ibs. 
161744 
145421 
136326 
110387 
94520 
69558 
51534 

Ft.  Ibs. 
132475 
111527 
99084 
67363 
53065 
18517 
8264 

Per  cent  . 
81.9 
76.69 
72.68 
61.02 
56.14 
26.62 
16.04 

0.7301 
0.7831 
0.687 
0.7508 
0.6702 
0.8073 
0.7225 

The  velocity — about  3.458  feet — in  the  supply  pipe  was  neg- 
lected in  determining  the  head,  for  which,  if  a  correction  be  ap- 
plied, it  will  reduce  the  brake  efficiency  to  about  80.59,  and 
hence  the  hydraulic  efficiency  would  be  about  82  per  cent. 

68.  The  Swain  Turbine. — The  Swain  turbine  is  a  mixed  now 
wheel,  inward  and  downward,  of  which  a  sectional  view  is  shown 
in  Fig.  45,  and  an  external  view  with  vertical  shaft  in  Fig.  44.  TF, 
Fig.  45,  is  the  wheel,  A  the  guides  which  are  secured  to  the  gate  6r, 
and  are  raised  and  lowered  with  it,  and  pass  into  and  out  of  the 
chamber  E.  By  this  arrangement  the  inner  ends  of  the  guides  are 
brought  nearer  the  buckets  than  when  the  gate  is  between  the 
bucket  and  guides,  the  space  in  this  wheel  being  If  inches.  The 
gate  is  opened  by  being  lowered,  so  that  the  water  first  enters  just 
under  the  crown,  and  passing  inward  and  downward  is  discharged 
nearly  radially,  while  that  which  enters  lower  is  discharged  nearly 
axially.  The  guides  and  upper  ends  of  the  buckets  are  shown 
in  Fig.  46.  There  are  three  heavy  cast-iron  guides,  one  of  which 
is  shown  in  Fig.  46,  through  which  pass  the  lifting  rods,  as 
shown  at  e.  The  other  21  guides  were  0.23  inch  thick  of  bronze, 
sharpened  at  the  ends  to  0.04  of  an  inch  thick,  nearly  19  inches 
long.  It  had  25  buckets  of  bronze,  pressed  into  shape  and  cast 


104:  HYDRAULIC    MOTORS. 

into  a  crown  above  and  band  below,  and  were  23.35  inches  deep. 
The  gate  G  was  made  of  two  cylinders  M  and  JV,  joined  by  a 
disc,  Q.  At  the  lower  end  of  M  is  a  narrow  flange,  to  which  was 
attached  a  leather  packing  to  prevent  the  escape  of  water.  The 
leakage  was  so  small  as  to  be  unimportant.  The  wheel  rested  on 
an  oak  pivot,  S,  conical  at  its  ends,  free  to  turn  or  rest,  and  sup- 
plied with  water  through  the  pipe  f.  It  was  so  arranged  be- 
tween the  connecting  piece  a  and  crown  Tr,  that  the  step  S  could 
be  replaced  by  another,  and  the  adjustment  of  the  height  of  the 
wheel  made  by  means  of  the  screws  #,  t.  Opposite  the  thick 
vanes  are  stationary  supports,  of  which  one  is  shown  at  0,  Fig. 
45,  resting  on  the  cast-iron  base  C,  and  support  the  chamber  E 
and  cover  of  the  wheel  L.  The  openings  in  the  base  C  for  the 
passage  of  water  into  the  wheel  pit  flow  outward,  as  shown  in 
the  right-hand  part  of  the  figure. 

An  elaborate  report  of  the  test  of  this  turbine — designed  to 
replace  the  centre-vent  turbine  established  in  1849  at  the  Boott 
Cotton  Mills,  previously  described — is  given  by  Francis  and 
published  in  the  Journal  of  the  Franklin  Institute  for  April, 
1875,  to  which  we  are,  through  the  courtesy  of  the  Swain  Manu- 
facturing Co.,  indebted  for  the  facts  and  tables  herein  given. 
Table  XXIII.  contains  only  the  best  result  for  each  gate  opening 
up  to  "full  gate."  Mr.  Francis  states  that  Table  XXIV.  is 
made  from  a  curve  of  the  results  plotted  on  section  paper. 

It  will  be  seen  that  the  highest  efficiency  was  obtained  with 
the  gate  closed,  1.08  inches.  This  indicates  that  there  may  have 
been  interferences  of  the  stream  of  water  in  the  Bucket  at  full 
gate  opening.  It  is  also  worthy  of  special  note  that  the  effi- 
ciencies are  well  maintained  down  to  the  smallest  opening  of 
2  inches.  The  hydraulic  efficiency  for  the  best  test  was  about 
85  or  86  per  cent. 

DIMENSIONS. 

Outer  diameter,       .     .     .     .     .     .     2/\  =  Oft. 

Least  inner  diameter,  .     .     ...     2/'2  =  2f  ft. 


FIG.  42. 


FIG.  48. 


HYDRAULIC    MOTORS. 

Depth  of  guide  passages,      .     .     . 

"         buckets  at  inflow,       .     . 

Mean  terminal  angle  of  guides,     . 

u      initial  angle  of  buckets,  .     . 

"      terminal   angle   of    buckets, 

estimated,       .     .     . 
Number  of  buckets,     .     .... 
a  guides,       .     .     .     .     , 

Thickness  of  guides,    .     .     .    -.     . 

Measured  area  between  guides, 
Least      measured     area     of      the 

buckets,  .     ...     .     .     .     .     /,-/    =  9.558  "     " 

Where  the  water  leaves  the  wheel  axially  (outer  diameter), 
y<2  =  26°,  and  where  it  leaves  it  radially,  ;/2  =  22°,  so  it  is  esti- 
mated that  the  effective  angle  is  about  25°. 

TABLE  XXIII. 

ABSTRACT   OF   THE   RESULTS  OF   EXPERIMENTS   WITH    "  SWAIN"  TURBINE 
WHEN  Rux  FOR  BEST  EFFECT. 


T  = 

13.08  in. 

y*   = 

13.285  in. 

a    = 

25° 

Yi    = 

90° 

7,    = 

25° 

^  = 

25 

ti  = 

0.23  in.  to 

t  = 

0.04  in.  at  end. 

K'  = 

9.880  sq.  -ft. 

OPENING 
OF  GATE. 
IN. 

No.  of 
rev.  per 
minute. 
n. 

Head. 
Feet. 
H 

Vcl  Water 
passing  weir 
per  second. 
Cu.  ft. 
Q. 

Energy  of 
the  fall 
Ft.  Ibs. 
&  QH. 

Brake 
Power 
Ft.  Ibs. 
U. 

Efficiency 
per  cent. 
E. 

Ratio. 

?*2    W1 

v*~g~H 

2 

60  5 

14  2S3 

51  177 

45518 

21572.5 

47.39 

0.6274 

3 

60.3 

13  977 

68  009 

59189 

34806.5 

58.81 

0.6623 

4 

60.0 

13.699 

83.894 

71562 

47672.5 

66.02 

0.6478 

5 

60  12 

13.482 

97  736 

82048 

59179.7 

72.13 

0.6537 

6 

60  21 

13  281 

110  092 

91044 

6.2263.5 

76.08 

0.6689 

7 

66.33 

13  102 

119.268 

97309 

76932.3 

79.06 

0.7184 

8 

60.40 

12  968 

130.253 

105184 

84794  3 

80.61 

0.6979 

9 

60  5 

12.836 

138.839 

110956 

91961.2 

82.88 

0.7098 

10 

60.43 

12  680 

144.774 

134202 

95583.9 

83  63 

0.7082 

11 

60  56 

12.640 

151.501 

139226 

99206.1 

83  26 

0.7227 

12 

67  6 

12  581 

156.703 

122158 

102435.2 

83.85 

0.7490 

13.08 

108  ..36 

13.099 

120  392 

98204 

20237.7 

20  61 

1.1738 

Full  Gate. 

97  8 

12  880 

137.871 

110580 

53189.4 

49.91 

1.0683 

87.7 

12.603 

149.133 

117041 

80804.4 

69.04 

0.9683 

78.0 

12.480 

158.877 

123471 

99109.8 

80.27 

0.8660 

72.7 

12.432 

161  225 

124814 

103522.0 

82.94 

0.8083 

69.1 

12.172 

162.538 

125223 

104652  9 

83.57 

0.7701 

1 

68.0 

12.408 

163.572 

126:387 

105401.9 

83  40 

0.7588 

' 

66.5 

12  361 

163.607 

125986 

104857.4 

83.23 

0  7412 

IOC 


HYDRAULIC    MOTORS. 


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The  Swain  Turbine 
FIG.'  44 


FIG.  44a. 


HYDRAULIC    MOTORS.  107 

69.  The  Hercules. — The  Hercules  turbine  is  also  a  mixed  flow 
wheel,  inward  and  downward.  The  entrance  surface  A  A,  Fig.  47, 
is  conical,  and  divided  horizontally  by  several  partitions,  so  as  to 
be  equivalent  to  several  wheels  superimposed, but  cast  and  working 
as  one.  The  partitions  are  for  the  purpose  of  securing  a  good  effi- 
ciency with  partial  gate  opening  ;  indeed,  the  best  efficiency  was 
for  partial  opening,  from  which  it  is  inferred  that  at  full  opening 
the  streams  within  the  wheel  interfered  with  each  other,  and 
so  prevented  their  producing  the  best  effect.  The  passages  are 
so  curved  that  the  inner  filaments  passed  downward  and  issued 
nearly  radially,  while  the  outer  ones  escaped  at  £  more  nearly 
axially.  A  cylindrical  gate  surrounded  the  wheel  and  was  opened 
by  being  raised,  similar  to  the  Fourneyron  style,  or  as  was  done 
in  the  Francis  wheels,  Figs.  19  and  31.  The  guides  were  outside 
the  gate  and  stationary,  as  in  Fig.  31,  and  divided  by  hori- 
zontal partitions.  In  1883  a  "Hercules"  was  tested  at  the 
Holyoke  testing  station.  The  wheel  was  of  the  following 
dimensions : 

DIMENSIONS  OF  THE  "  HERCULES"  TURBINE. 

Mean  external  radius, %rm  =  36  in. 

Terminal  angle  of  centre  line  guides,  a     =  14f  ° 

Initial  angle  of  buckets,       .     .     .     .  y\    —  98° 

Terminal  angle  of  buckets  (say)    .     .  /2     —  14° 

Measured  least  section  of  guides,       .  K'  —  4.752  sq.  ft. 

"  "     terminal     section    of 

buckets,      .     .     .... A-a'    =  7.925  sq.  ft. 

Number  of  buckets, N    =  24 

"         "  guides, N,   =  17 

The  first  test  showed  such  remarkable  results  in  regard  to  effi- 
ciency, that  it  was  followed  by  two  other  tests,  all  of  which 
seemed  to  confirm  each  other.  The  following  are  from  the  third 
test: 


108 


HYDRAULIC    MOTOKS. 


TABLE    XXV. 
ABSTRACT  OP  (THIRD)  TEST  OF  A  GO-INCH  HERCULES  TURBINE. 


6 
• 

CO 

a 
|| 

I 

Id 

• 

1 

0 

o| 

t* 

u 

o  'o^ 

v  g; 

I* 

h" 

r*  1 

O 

d 

•g 

"o  O. 

« 

s  ~ 

fc 

* 

^ 

M 

1.000 

Still 

16.82 

0 

0 

0 

0 

129.10 

16.94 

89.66 

145.59 

84.55 

0.614 

135.33 

16.95 

89.00 

146.02 

85.38 

0.644 

140.62 

16.96 

88.33 

145.72 

85.80 

0.669 

144.80 

16.98 

87.79 

143.88 

85.14 

0.688 

0.806 

123.00 

16.88 

79.38 

129.71 

85.39 

0.586 

130.25 

16.88 

78.63 

131.01 

87.07 

0.621 

137.00 

16.92 

78.00 

130.28 

87.07 

0.652 

143.00 

16.94 

77.38 

129.01 

86.82 

0.680 

0.647 

126.50 

17.09 

67.25 

111.04 

85.22 

0.599 

134.67 

17.15 

66.52 

111.65 

86.33 

0.637 

140.40 

17.16 

65.72 

109.55 

85.69 

0.664 

0.489 

115.60 

17.25 

55.54 

85.27 

78.50 

0.545 

123.00 

17.22 

•     54.80 

85.48 

79.90 

0.581 

130.00 

17.28 

54.07 

84.79 

80.05 

0.613 

0.379 

112.75 

.17.61 

45.20 

65.29 

72.36 

0.526 

122.67 

17.65 

44.50 

65.06 

73.06 

0.572 

129.40 

17.66 

43.76 

63.89 

72.93 

0.603 

The  hydraulic  efficiency  was  probably  about  87  or  88  per  cent. 

Analysis  of  the  "  Hercules"  Wheel. — Making  rl  =  r^  = 
rm  =  18",  a  =  Uf,  Y,  =  98°,  y,  =  14°,  ^  =  0.10,  //3  =  0.05, 
gives 

E  =  O.8998  =  9O  per  cent,  nearly. 

With  the  same  data  except  /*,  =  ^  =  0.10  gives 
E—  O.88O3  =  88  percent., 

which  exceeds  the  brake  efficiency  only  one  per  cent.,  and  indi- 
cates that  the  resistances  are  low  and  about  0.10  for  //,  and  /v 


FIG.  46 


HYDRAULIC    MOTORS.  109 

70.  The  Victor.— The  "  Victor"  turbine  followed  the  Her- 
cules in  its  introduction  to  the  public.  It  is  a  mixed-flow  tur- 
bine, the  water  entering  radially  inward  at  the  circumference, 
thence  discharging  downward  and  outward.  The  whole  body  of 
the  wheel,  except  that  for  the  shaft  and  step,  is  occupied  by  the 
buckets,  which  are  deep  axially,  thus  giving  great  capacity  for  its 
size. 

The  water  is  regulated  by  two  styles  of  gates :  one  is  called 
"  the  register  gate ;"  the  other,  "  the  cylindrical  gate."  The 
former  opens  the  passages  for  the  water  by  turning  about  the 
axis  of  the  wheel,  thus  opening  the  passages  their  whole  length 
and  making  the  opening  wider  and  wider  as  it  is  turned  more 
and  more,  and  at  the  same  time  gives  direction  to  the  wrater  ;  the 
latter  is  a  cylinder  moving  axially>  and  opens  the  passages  their 
full  width  as  the  gate  is  raised.  The  latter  is  preferable  when 
the  water  supply  is  variable  and  not  sufficient  to  fill  the  passages 
at  full  gate,  or  when  the  work  is  variable  ;  for  a  better  efficiency 
is  obtained  with  it  at  "  partial  gate."  A  view  of  the  latter  is 
shown  in  Fig.  48,  for  which  we  are  indebted  to  the  courtesy  of 
the  manufacturers,  Stillwell-Bierce  and  Smith- Yaile  Co.,  Day- 
ton, Ohio. 


110 


HYDRAULIC    MOTORS. 


TABLE  XXVI. 

RESULTS  OF  TESTS  OF  THE  "VICTOR"  TURBINE  AT  THE  HOLYOKE  TESTING 
STATION,  WITH  THE  "CYLINDER  GATE." 


SIZE  OF  WHEEL  AND 
GATE  OPENING. 

Head 
in  Feet. 
H. 

Revolutions 
of  Wheel 
per  minute. 
It, 

Cubic  Feet 
Water 
per  minute. 
60  Q. 

Horse-power 
Developed 
by  Wheel. 
H.P. 

Percentage 
Tseful 
Effect. 
E. 

30-inch  Full  Gate.     .  . 

17.51 

168 

4440 

119.56 

81.35 

4 

17.82 

163 

3892 

104.93 

80.03 

4 

17.95 

163 

3392 

88.24 

76.66 

f        "     

18.10 

155 

2893 

70.97 

71.28 

•J-        " 

18.20 

159 

2265 

51.42 

63.46 

36-inch  Full  Gate 

16.78 

135 

6106 

158.18 

81.80 

1        " 

17.14 

135 

5422 

141.58 

80.71 

£ 

17.35 

140 

4708 

118.22 

76.68 

4c                            

4                " 

17.05 

129 

3982 

91.62 

71.50 

4         "     

17.48 

134 

3202 

66.87 

63  30 

39-inch  Full  Gate  

14.66 

116 

6873 

152.66 

80.37 

1        "     .   ... 

14.53 

118 

5920 

129.41 

79.80 

1         "     

16.84 

125 

5517 

135.56 

77.40 

i        |'     

17.06 

123 

4695 

108.22 

71.67 

17.39 

124 

3856 

81.00 

64.07 

48-inch  Full  Gate  

13.23 

91 

10072 

201.71 

80.11 

i          \    

14.36 

89 

9042 

192.41 

78.42 

14.75 

89 

7869 

165.23 

75.34 

I        "     ."'.! 

14.87 

85 

6744 

132.76 

70.06 

i        "    

15.28 

87 

5526 

100.66 

63.09 

NOTE.— In  the  above  calculations  "fractional  gate"  means  "fractional  water,11  regardless 
of  the  position  of  the  gate  of  the  turbine  ;  that  is.  "  three-quarters  gate'1  means  that  the  gate 
i»  closed  to  a  point  where  the  discharge  of  water  is  only  three-fourths  of  what  it  is  at  full  gate 
tinder  the  same  head.  This  is  nearly  but  not  perfectly  exact, 


THE  VICTOR  TURBINE* 

11  CYLINDER   GATE" 
FIG.  46. 


HYDRAULIC    MOTORS. 


Ill 


TABLE   XXVII. 

RESULTS  OF  TESTS  OF  THE  "  VICTOR  "  TURBINE  AT  THE  HOLYOKE  TESTING 
STATION,  WITH  THE  "  REGISTER  GATE." 


SIZE  OP  WHEEL. 
FULL  GATE. 

Head 
in  Feet. 
H. 

Revolutions 
per 
minute. 
n. 

Horse- 
power. 
H.P. 

Cubic  Feet 
of  Water 
per  minute. 
60  q. 

Percentage 
Useful 
Effect. 
E. 

15-inch              ] 

18.06 

368 

30.17 

990.19 

.8932 

17}  inch             •) 

18.08 
18.02 

355 

280 

30.  12 
35.51 

996  .  83 
1164.60 

.8849 
.8960 

20-inch         j 

17.96 

18.22 

292 

286 

36.35 

48.75 

1197 
1660.17 

.8950 
.8532 

25-inch  •] 

18.23 
17.79 

275 
205.5 

48  .  75 

67.72 

1660.17 
2362.72 

.8528 
.8530 

30-inch  •] 

7.96 
11.65 

209 
144.5 

68.62 
52.54 

2356  .  54 

2751.87 

.8584 
.8676 

35-inch      ...          -j 

.DO 

17.31 

147.5 
151.7 

ol  .  yo 
135.68 

27o5  .  09 
4895 

.  8564 

.8489 

40-inch      •] 

i  .29 
16.49 

160 
130 

133.19 
148.93 

4806 
5789 

fJQ1  R 

.8497 
.8253 

44-inch  

lO.4< 

15.50 

109.25 

14o.oo 

155.78 

Oolo 
6643 

.OZ4L 

.8003 

48-inch    

15.51 

102 

179.29 

7456 

.8202 

It  will  be  seen  that  the  highest  efficiencies,  over  89  per  cent., 
border  very  closely  upon  the  figure  which  many  writers  reject 
without  discussion.  The  engineer  who  made  the  tests  has  re- 
marked, in  effect,  that  the  smallest  wheels  were  too  small  for 
determining  the  efficiency  with  great  accuracy.  Even  if  these 
be  excluded,  the  efficiencies  are  high. 

71.  Pelton  Wheel. — The  Peltoii  wheel,  or,  as  it  was  formerly 
called,  the  "  hurdy-gurdy"  wheel,  has  been  popular,  especially  in 
California,  for  utilizing  the  power  of  high  waterfalls.  It  con- 


112  HYDRAULIC    MOTORS. 

sists  of  a  series  of  curved  double  buckets,  as  shown  in  Figs.  48 
and  49,  attached  to  the  circumference  of  the  wheel.  A  is  the 
nozzle,  B  the  valve  case,  in  Fig.  49# .  The  water  impinges  against 
the  buckets  at  the  partition  and  flows  outward  in  opposite  direc- 
tions, discharging  at  the  outside  planes  of  the  wheel.  It  is  then 
an  impulse  wheel,  or  wheel  of  free  deviation,  with  axial  flow  and 
segmental  feed,  and  may  be  analyzed  according  to  Article  30. 
For  the  limiting  values  of  the  angles,  we  have 

a  =  0,  Yl  =  180,  y,  =  0  ;  r,  =  r,  ;  ^  =  ^  =  0  ; 
for  which  we  find 


GO  i\  =  velocity  of  circumference  =  -J   4/2  g  H, 

equation  (36),  which  is  half  the  velocity  due  to  the  head.  The 
velocity  of  discharge  will  be,  equation  (34),  F~2  =  0,  and  the 
efficiency  will  be  unity.  But  these  are  impossible  conditions  in 
practice.  The  angle  EEC  must  be  less  than  90°,  so  that  the 
water  will  escape  before  it  is  struck  by  the  succeeding  bucket. 
Draw  B  D  parallel  to  the  line  of  the  jet  (or  in  the  plane  of  the 
wheel),  to  represent  the  velocity  of  the  wheel,  and  B  C  tangent  to 
the  bucket  at  its  terminus,  to  represent  the  velocity  of  the  water 
as  it  leaves  the  bucket  relative  to  the  bucket,  and  complete  the 
parallelogram  ;  then  will  EB  represent  the  velocity  of  exit  from 
the  bucket  relative  to  the  earth.  Considering  the  angle  at  A  as 
zer\>,  F,  the  velocity  of  the  rim  of  the  wheel  —  B  D  —  C  E, 
and  V  the  velocity  of  the  jet,  then  will  the  relative  velocity  of 
the  water  along  the  concave  surface  of  the  bucket  be 

V—  V,  =  B  C  =  ED  =  ^  V^JI—  V,. 

If  the  velocity  of  the  wheel  be  such  that  E  B  is  perpendicular 
to  the  jet  (or  to  the  plane  of  the  wheel),  then 


HURDY-GURDY     WHEEL. 

FIG.  48. 


FIG.  49. 


HYDRAULIC    MOTORS.  113 

V,  =  &  C  cos  E  C  B  =  (  V—  FO  cos  yf 
COS  - 


cos  yt 


y    _    £g  _   £  £  gin  _  SIR   K2  y- 

1  +  <^os  72 


Thus,  if  ;i  =  0.98,  EC  B  =  20°,  then 


F=  0.98   ^2,   Vl  =  0.4T6   F,   Fa  =  0.18  V,  E  =  0.928, 
or  nearly  93  per  cent. 

If  the  velocity  of  the  wheel  be  half  that  of  the  jet,  then 

EC=   V,  =  j-  F; 
BC  =  V    -  F,  =  i  F; 
.'.B  C  =  EC', 
.-.  F,  =  ^^  =  2  sin  J-  (7; 

and  £7  and  ^  the  same  as  that  given  above.  When  the  angle  0 
is  small  —  say  less  than  25°,  "2  sin  -J  C  =  sin  (7  with  sufficient 
accuracy  for  this  case,  and  the  results  will  be  very  nearly  the 
same  as  those  given  above,  so  that  for  the  conditions  assumed 
above,  the  theoretical  efficiency  will  be  about  93  per  cent,  for 
this  case.  The  efficiency  will  be  nearly  the  same  for  a  speed 
several  per  cent,  above  or  below  that  for  the  maximum. 

If  the  angle  C  —  30°,  and  the  half  angle  at  A  also  30°,  and 
the  speed  be  one-half  the  component  of  the  velocity  of  the  jet, 
and  p  =  0,96;  then 

F      =  0.96   4/2^7?; 

F,      =  i   F  cos  30°  =  0.433  F, 

B  C  =   F--F,  =  0.433  F, 

F2      =  2  B  C  sin  15°  =  0.224  F; 
.  •  .  E  =  0.92  (1  -  0.224*)  =  0.874. 


114:  HYDRAULIC    MOTORS. 

In  these  computations  no  allowance  lias  been  made  for  the  fric- 
tion in  the  feed  pipe,  nor  for  imperfect  working  at  the  junction 
of  the  buckets,  nor  for  loss  due  (if  any)  to  imperfect  discharge. 
These  all  operate  to  reduce  the  efficiency. 

In  a  paper  read  by  Mr.  Hamilton  Smith,  Jr.,  before  the  Ameri- 
can Society  of  Civil  Engineers,  February  6,  1884,  a  test  is  reported 
in  which  the  efficiency  was  given  as  87.3  per  cent,  with  a  speed 
of  0.51  t/2  y  //,  under  a  head  of  386  feet,  wheel  6  feet  in  diam- 
eter. He  states  that  the  wheel  carries  over  a  large  amount  of 
water.  This  efficiency  is  remarkable,  but  if  it  be  admitted  that 
there  were  incidental  errors  in  the  test,  it  still  shows  that  the 
efficiency  was  high,  and  Mr.  Hamilton  infers  that  it  was  certainly 
as  high  as  85  per  cent. 

A  small  "  hurdy-gurdy"  was  tested  at  Stevens'  Institute  by 
some  students  in  1891,  for  which  the  efficiency  was  found  to  be 
less  than  70  per  cent. 

The  wheel  is  especially  commendable  for  great  heads  :  it  is 
simple  in  construction,  durable,  efficient,  easily  managed,  and 
easily  repaired.  It  was  invented  by  a  village  carpenter,  who, 
after  reading  Francis'  Lowell  Hydraulic  Experiments,  and  resid- 
ing near  a  fall  of  water,  made  a  Prony  friction  brake  and  a  weir, 
and  by  continuous  experiments  arrived  at  the  particular  form  and 
setting  of  buckets  which  he  adopted  as  the  best.  The  juncture 
A  in  the  actual  bucket  is  back  of  the  line,  joining  the  extremities 
through  B. 

72.  Poncelet  Wheel. — The  Poncelet  wheel  is  the  invention  of 
him  whose  name  it  bears.  In  accordance  with  hydraulic  prin- 
ciples, the  inventor  changed  the  plane  float  wheel,  which  had  an 
efficiency  of  about  16  per  cent.>  to  one  with  curved  buckets,  Fig. 
50,  raising  the  efficiency  to  60  or  70  per  cent.  The  water  enters 
the  buckets  tangentially,  moving  up  the  concave  side  with  a  dimin- 
ishing velocity  since  it  works  against  gravity,  until  it  ceases  to  as- 
cend, when  it  descends  and  leaves  the  bucket  with  a  backward 


Fw.60. 


HYDRAULIC    MOTORS.  115 

velocity  in  reference  to  the  bucket,  its  velocity  in  reference  to  the 
earth  depending  upon  the  velocity  of  the  wheel.  The  energy  lost 
in  its  work  against  gravity  is  restored  by  gravity  during  descent, 
neglecting  friction  in  the  buckets,  so  that  it  is  only  necessary  to 
consider  the  energy  of  the  jet  at  entrance  and  quitting.  It  is 
virtually  a  turbine  of  "  free  deviation,"  with  segmental  feed. 
Considering  the  limiting  case,  in  which  the  terminus  of  the 
bucket  is  tangent  to  the  outer  circumference,  and  that  the  water 
enters  tangentially,  then 

a  —  0,  yl  —  180°,  y,  =  0,  /*,  =  0  —  /*„  r,  =  r» 
then 


Equation  (93),       V  =    V%  g  H, 
«        (94),      vl  =   V  --   w  r, 

(97),  G*  r  =  J   V^~H  =  i  F, 


" 


(101),      U=$M  F2, 
(102),      E  =  1. 


But  these  conditions  cannot  be  realized  in  practice.  The  ter- 
minal angle  of  the  bucket  has  an  angle  of  15°  or  more  with  the 
circumference,  or  yl  —  165°.  The  angle  of  the  guide,  or  chute, 
will  be  20°  or  25°  or  even  more,  and  y^  =  15°  or  more,  and 
yu1  =  0.10  or  more.  These  would  give  an  efficiency  of  less  than 
90  per  cent.  There  may  be  a  further  loss  by  water  escaping  be- 
tween the  wheel  and  apron  below  the  wheel,  and  there  may  be 
an  imperfect  action  of  a  finite  stream  ;  since  the  bucket  may  strike 
the  stream,  and  the  stream  strike  the  crown,  thus  reducing  the 
theoretical  efficiency.  Actual  tests  give 

E  =  0.55  to  0.70, 
when 

v  =  0.55  F  about. 


116  HYDRAULIC    MOTORS. 

73.  In  the  preceding  tests  we  have  given  the  highest  effi- 
ciencies, but  it  should  not  be  inferred  that  these  high  figures  are 
attained  in  the  majority  of  cases.  In  Europe,  Rittenger  tested 
eight  turbines,  parallel  flow,  mean  radii  from  6  to  11  inches, 
having  various  vane  angles,  under  heads  varying  from  about  6  to 
18  feet,  for  which  he  found  efficiencies  varying  from  0.63  to  0.71. 
It  may  be  observed  that  tests  of  American  wheels  in  America— 
or  we  might  say  in  Massachusetts — have  given  higher  efficiencies 
than  tests  in  Europe,  and  the  question  has  been  raised,  if  the 
difference  may  not  be  due  to  the  different  methods  of  measuring 
the  quantity  of  water ;  but  the  difference  is  not  accounted  for  in 
this  way.  There  appears  to  be  essential  differences  in  the  wheels 
in  favor  of  the  American  types. 

In  designing,  it  is  not  advisable  to  use  the  highest  attainable 
figures  for  the  efficiency,  for  when  tested  the  wheels  are  supposed 
to  be  in  their  best  condition,  well  lubricated  and  bearing  perfect, 
and  after  long  service,  some  parts  may  be  so  worn  as  to  make 
the  wheel  less  efficient ;  hence  70  per  cent.,  or  certainly  not  to 
exceed  75  per  cent.,  should  be  assumed.  In  ordinary  practice, 
with  variations  of  speed  above  or  below  that  which  would  pro- 
duce a  maximum,  with  wheels  not  constructed  with  great  care, 
and  with  a  lack  of  proper  attendance  they  may  fall  below  60  per 
cent. 


HYDRAULIC    MOTORS.  117 


CLASS-ROOM  EXERCISE. 

(The  following  exercise  was  conducted  by  the  author  with  a 
class.) 

Design  a  turbine  to  utilize  the  power  of  a  stream  having  an 
available  fall  of  16  feet. 

74.   To  find  the  power  of  the  fall. 
Let 

§  be  the  weight  of  a  cubic  foot  of  water, 
Q,  the  volume  of  water  falling  per  second, 
H,  the  height  of  the  fall ; 
then  the  energy  of  the  fall  per  second  will  be 

SQH, 

and  the  horse-power, 

77   P  _  *  Q  H 
550    ' 

The  weight  of  a  cubic  foot  of  water  depends  upon  its  tem- 
perature, latitude  and  elevation  of  the  place  ;  but  in  the  use  of 
water  wheels  the  temperature  will  vary  so  little  from  60°  or  70°, 
and  the  latitude  and  elevation  will  have  so  small  an  effect,  we 
will  use 

tf  =  62.4.* 

*  If  T  be  temperature,  degrees  Fah. , 
"  I,  the  latitude  of  the  place, 
"  e,  the  elevation  above  sea  level, 

"  60  =  62,375,  the  weight  of  a  cubic  foot  of  water  at  its  maximum  den- 
sity, 39.1°,  at  the  level  of  the  sea  at  latitude  45°,  then 

6  =  60  (1  -  0.0002)  (T-  39.1°)  (1  -  0.000256  cos  2l)(l  —  ?^ 

If  r  be  the  radius  of  the  earth  at  the  place, 
rfl     "  '"  "  "  45°,  then 

r   —  To  (i  +  0.00164  cos  2  I), 
r0  =  20,892,200  feet. 


118  HYDRAULIC    MOTORS. 

In  case  of  accurate  tests,  the  proper  allowance  for  these  changes 
may  be  made. 

The  head  over  the  crest  of  the  weir  will  vary  from  zero  at  the 
crest  to  some  finite  value  depending  upon  the  depth  of  water 
over  the  crest.  To  deduce  a  formula  for  this  case,  first  assume 
that  the  water  flows  through  an  orifice  B  D,  Fig.  51,  in  which 
C  D  is  the  breadth,  B  C  the  depth,  and  A  the  free  surface.  Let 
E  be  a  rectangular  element  =  d  x  d  y,  in  which  y  is  vertical 
and  x  horizontal,  and  y  the  depth  of  E  below  the  free  surface  ; 
then 

d  Q  =  ft  V%  g  y  ,  d  y  d  x. 
Let 

I*  be  a  coefficient  of  discharge, 

A2,  the  depth  C  A, 
A,,   «        «      BA, 

I,  the  breadth  C  D  ; 
then 


Q  =  v  VZ  g  \'y  dydx  =      ^   V^gl     V    -  *•   O) 

If  the  upper  surface  of  the  orifice  be  free  —  in  other  words,  if 
the  orifice  be  a  weir,  as  in  Fig.  52  —  it  has  been  found  by  experi- 
ment that  Aj  may  be  taken  as  zero  when  7?2  =  B  D,  measured 
from  the  level  A  B  of  the  water  several  feet  above  the  weir 
down  to  the  crest  of  the  weir  D.  The  coefficient  ^  varies  be- 
tween 0.60  and  0.64,  depending  upon  the  depth  C  D  over  the 
weir  and  B  F  the  breadth  ;  and  if  one  were  gauging  a  stream 
where  great  accuracy  was  essential,  it  would  be  necessary  to  re- 
sort to  tables  to  determine  the  exact  coefficient  to  be  used 
(D'Aubisson,  Hydraulics,  or  Weisbach,  Hydraulics),  or  resort  to 
Francis'  formula  as  given  in  Lowell  Hydraulic  Experiments, 
which  is 

Q  =  3.33  (I  —  0.1  n  h)  h%, 


FIG.  51- 


B 


1 

1  // 


l&&m*£tt 


^^:^M^m 


F 

3T 


FIGK  52. 


HYDRAULIC    MOTORS.  119 

in  which  n  is  the  number  of  contractions  (two,  if  both  ends  are 
contracted  ;  one,  if  only  one  end  ;  and  zero  is  neither  end). 
If 

n  =  0, 

Q  =  3.33  I  $, 

to  which  the  preceding  formula  will  reduce  if  k1  =  0  and 
fj.  =  0.622.  Streams  vary  continually  in  the  quantity  of  water 
discharged,  so  we  will  use  the  smaller  coefficient,  or  /*  =  0.60, 
and  our  equation  becomes 

Q  =  3.21  I  A*  (J) 

Let  a  weir  be  constructed  of  boards  having  bevelled  edges — the 
sharp  edges  being  placed  up  stream — and  by  measurements  sup- 
pose it  be  found  that 

I    =  6  feet, 
/<„  =  1.83  feet. 

The  height  of  the  surface  may  be  accurately  found  by  a 
"  hook-gauge,''  a  device  invented  by  Mr.  Francis,  in  which  a 
hook  is  submerged  and  then  gradually  raised  until  the  point  is 
just  visible  in  the  surface.  In  this  way  the  height  of  the  surface 
.may  be  found  with  much  accuracy  to  a  small  fraction  of  an  inch. 
We  have 

Q  =  3.21  X  6  X  (1.83)  *  =  48  cubic  feet 

per  second,  nearly,  and  sufficiently  accurate  for  our  present  pur- 
pose. The  theoretical  power  of  the  fall  will  be 

H.  P.  =  4S  X  62'4  X  16  =  87.13 

550 

horse-power.  If  the  turbine  has  75  per  cent,  hydraulic  effi- 
ciency, it  will  have  65.35  horse-power  ;  and  if  the  frictional  re- 
sistances be  3  per  cent,  of  this  power,  it  will  have  at  the  brake 
63  horse-power.  From  this  we  can  judge  whether  the  stream 
will  supply  the  required  power. 


120  HYDRAULIC    MOTORS. 

75.  To  determine  the  diameter  of  the  ivheel.  —  We  choose  for 
this  exercise  a  radially  outflow  wheel.  The  greater  the  diameter 
of  the  wheel,  the  less  will  be  its  depth  in  order  to  discharge  the 
given  amount  of  water.  There  is  no  recognized  rule  for  deter- 
mining the  diameter.  The  depth  between  the  crowns  of  the  Tre- 
inont  wheel  was  about  \  the  outside  diameter.  If  proportioned 
for  space  or  to  economize  material,  the  depth  would  be  more 
nearly  equal  to  the  radius.  We  will  try  a  depth  equal  to  J  the 
radius,  rv  Table  VI.,  column  10,  second  case,  gives 

F22  =  0.04  X  2  g  H. 
.'.  F2  =  0.2  1/2  g  H. 

The  Tremont  wheel  gave 


F2  =  0.24  VY^H  =  V^YH,  nearly. 

Assuming  that  the  free  opening  at  the  outer  rim  (the  circum- 
ference less  the  space  occupied  by  the  walls  of  the  buckets)  is 
•f  of  the  outer  circumference,  and  that  the  radial  velocity  at  quit- 
ting is  0.2  1/2  g  H,  and  depth  J  r»  we  have 


|  .  2  n  r,  .  J  r,  .  0.2  1/2  g  H  =  48  ; 
.  • .  /*a  =  2.6  feet,  nearly 
=  31.2  inches. 

The  object  of  this  investigation  is  not  to  fix  exactly  the  pro- 
portions of  the  wheel,  but  to  determine  approximately  such  pro- 
portions as  may  be  previously  desired.  We  may  now  assume, 
arbitrarily,  the  outer  radius,  and  if  not  satisfied  with  the  final  re- 
sult, recompute  with  another  assumed  radiiis. 

We  observe  that  the  smaller  the  diameter,  the  greater  will  be 
the  number  of  revolutions  per  minute,  and  the  wheel  may  be 
proportioned  to  give  the  required  revolutions ;  but  the  analysis 


HYDRAULIC    MOTORS.  121 

is  complex,  as  shown  by  equations  (15#)  and   (16),  page  8.     By 
the  aid  of  Table  VI.,  a  first  approximation  may  be  made ;  thus,  if 

n  be  the  number  of  revolutions  per  minute,  then 


„,/ 

{JeJ      — 


n  .  2  n       n 


60        "30' 
and  from  Table  VI. 

rt  =  0.8T5 

Thus,  if  II  =  16  feet,  n  =  100  per  minute  ;  then 
/•„  =  2.67  feet  =  32.04  inches ; 

but  the  coefficient  0.875  depends  upon  values  we  have  not  yet 
fixed.  In  several  cases  given  in  the  preceding  pages,  the  velocity 
of  the  initial  rim  to  the  velocity  due  to  the  head  is  between  0.62 
and  0.70.  We  now  assume  a  value  near  those  found  above,  but 
otherwise  arbitrarily, 

/>2  =  30  inches. 

The  width  of  the  crown  will  now  depend  upon  the  inner 
radius,  and  no  rule  exists  for  determining  this.  In  the  Trernont 
wheel  ?'„/?•,  =  1.23.  Rankine  gives  r^  —  i\  |/2  =  1.41  rr  These 
give,  for  our  problem,  1\  =  24.4  inches  and  21.27  inches  respec- 
tively ;  and  for  width  of  crown  5.6  and  8.73  inches  respectively. 
The  analysis  on  page  26  shows  that,  theoretically,  the  crown 
should  be  comparatively  narrow  for  the  outflow  wheel,  but  from 
physical  considerations  the  crown  should  be  sufficiently  wide  to 
secure  the  full  effect  of  the  stream  ;  but  if  too  wide,  friction  and 
difficulty  in  proper  feeding  might  be  prejudicial.  For  the  pur- 
pose of  study,  a  wide  crown  will  show  more  clearly  all  the  pecu- 
liarities of  the  buckets  than  a  very  narrow  one,  especially  in  the 
graphical  construction.  Then  assume  r^  =  20  inches,  giving  10 
inches  for  the  width  of  crown.  (The  width  of  crown  in  the 
Tremont  was  nearly  9J  inches.) 


122  HYDRAULIC   MOTORS. 

76.  Initial  angle  of  the  bucket.  —  For  reasons  given  previously, 
we  make 

Xl  =  90°. 

77.  Terminal  angle  of  bucket.  —  The  discussion  in  Article  5, 
page  9,  shows  that  y9  should  be  small  for  high  efficiency.     The 
smaller  it  is  the  greater  must  be  the  outer  depth,  and  if  the  wheel 
is  to  be  cast,  the  angle  should  not  be  too  small.     In  the  Tremont 
wheel,  in  which  the  walls  of  the  buckets  were  of  Russia  sheet 
iron  and  fitted  into  place  by  special  tools,  this  angle  was  10  de- 
grees.     Other  wheels  mentioned  in  this  work  are  made  with 
larger  angles.     We  will  assume 


78.  Terminal  angle  of   the  guide  vanes,   directrices,   or  dis- 
tributors. —  The  proper  value   of  a  is  discussed  on  page  16.  and 
that  the  internal  pressure  at  the  gate  should  exceed  that  of  an 
atmosphere  it  should  be  less  than  45°  for  yl  =  90°.     The  smaller 
it  is  the  higher  the  efficiency,  although  the  gain  in  efficiency  is 
small  per  degree  of  decrease  of  the  angle.     It  may  be  seen  in 
Table  X.,  that  for  an  increase  of  a  from   21°  to  23°,  the  loss  of 
efficiency  was  0.23  of  one  per  cent.,  or  not  far  from  y1^  of  one 
per  cent,  per  degree  for  those  values.     We  will  assume  the  fairly 
practical  v?]ue, 

a  =  30°. 

79.  Buckets.  —  There  is  no  recognized   rule  for   determining 
the  number  of  buckets  or  their    form.      Francis'  rule  for  the 
number  is 

^=3  (D  +2), 

where  D  is  the  outside  diameter  in  feet  ;  but  he  did  not  follow 
it  in  the  construction  of  his  wheel.  He  made  N  =  44,  while 
the  rule  would  give  30.  In  our  example  it  gives  13£,  and  12  or 
14  would  be  used  ;  or  if  in  the  proportion  of  the  diameters  to  the 
Tremont,  about  26.  The  wheels  tested  at  the  Centennial  Expo- 


HYDRAULIC    MOTORS.  123 

sition.liad  less  buckets,  page  41,  than  given  by  this  formula.  Cir- 
cumstances pertaining  to  the  ease  and  certainty  of  construction, 
or  of  obstacles  entering  the  wheel,  may  have  controlling  influ- 
ences. A  case  is  related  of  a  wheel  choked  and  stopped  by  eels, 
but  it  is  rare  for  a  wheel  to  act  as  an  eel-trap  ;  but  if  there  is 
^danger  of  obstructions,  it  would  be  wise  to  make  the  buckets 
larger.  We  will  try  ^"=18,  and  16  guides.  The  thickness  of 
the  partitions  will  depend  upon  the  mode  of  manufacture.  If 
of  sheet  steel,  they  may  be  -J  inch  thick  ;  if  of  bronze,  less  than  J 
of  an  inch ;  but  if  of  cast  iron,  they  should  be  thicker  to  secure 
sound  castings,  and  some  allowance  may  be  made  for  imperfect 
setting  of  the  cores.  So  many  cores  will  be  necessary  that  dry 
sand  moulds  should  be  used,  in  which  case  sound  castings  may  be 
secured  if  f  inch  thick  and  possibly  if  less ;  but  to  provide  for 
contingencies,  we  will  design  them  T7g  inch  thick  and  bevel  the 
initial  ends  to  a  sharp  edge.  With  this  data  we  make  the  first 
calculation.  The  data  are 

rt  =  20",  x,  =  90°>  A«i  =  °-10>  H  =  10  ft- 

n  =  30",  y.,  =  14°,  yu2  =  0.20,   Q  =  48  cu.  ft.  per  sec. 

t   =  Ty,  a    =  30°,  lf=lS  buckets. 


RESULTS. 

M  2  = 

0. 

608 

A7* 

— 

0.052 

Ang. 

vel.  GO'  = 

10. 

92 

E 

= 

0.737 

Rev.  = 

1.738 

per  sec. 

K 

= 

0.127 

sq. 

ft. 

— 

104. 

28 

per  min. 

k, 

= 

0.254 

a 

a 

F  = 

21. 

016 

ft. 

&2 

~ 

0.089 

u 

a 

vl  — 

10. 

501 

a 

2/i 

=B 

0.466 

ft. 

v9  =     29.824 

a 

2/2 

1=. 

0.511 

" 

a,  = 

0. 

545 

a 

e 

=  77°  10' 

0,  = 

0. 

722 

a 

H  P 

= 

64.31, 

hydraulic 

F,= 

f-r 
i. 

403 

u 

horse 

-power  of 

the  wheel. 

80.  Form  of  the  bucket. — If  the  bucket  is  to  be  described  with 
several  arcs  of  circles,  proceed  as  in   Fig.  19.     Let  F,  Fig.  19 


12-t  HYDRAULIC    MOTORS. 

or  53  be  the  terminus  of  a  bucket,  draw  the  radius  F  0,  and 
lay  off  F  c,  making  an  angle  of  14°  with  F  0.  Choose  a  con- 
venient point  <?,  on  F  c,  within  but  not  far  from  the  inner  arc  of 
the  crown  for  a  centre,  and  with  c^  F  as  a  radius,  describe  an  arc, 
F  fv  over  about  one  fourth  the  width  of  the  crown  ;  then  with 
a  shorter  radius,  /*,  <?„  where  <?2  may  be  still  within  the  inner  arc 
of  the  crown,  describe  /",  /*, ;  then  with  c3  the  arc  f3  ft ;  then 
with  c'4  complete  the  arc.  If  the  initial  angle  yl  be  90°,  the 
centre  <?4  must  be  on  the  tangent  to  the  inner  circumference  at 
G,  and  if  it  does  not  come  out  that  way  the  correction  should  be 
made  by  trial.  The  arcs  through/!,,  /*„  and/*4,  with  O  as  a  centre, 
may  first  be  described  dividing  the  crowns  into  four  equal  widths, 
or  any  other  proportion  if  desired. 

It  was  found  in  the  study  of  the  Tremont  turbine  that  if  the 
vane  angles  were  used  in  the  formulas  and  the  depth  ya  be  com- 
puted from  the  equation 

(2  n  •/•„  sin  y,  —  N  t)  v,  y2  =  Q, 

that  ?/2  was  found  to  be  too  large.  To  find  a  value  more  nearly 
the  practical  one,  draw  the  tangent  a  &',  Fig.  53,  and  through  u 
the  middle  of  the  arc  F  d,  a  perpendicular  p  r  to  Fg,  it  will  be 
found  that  p  is  a  point  of  tangency  of  a  line  parallel  to  Fg. 
The  terminal  angle  y^  at  u  will  be  the  same  as  the  vane  angle  at 
#,  or  14°  ;  hence  the  computed  mean  velocity  at  that  point  will 
be  v9  =  29.82.  Assume  0.85  as  the  coefficient  of  contraction, 
although  in  the  Tremont  wheel  it  was  0.88,  and  would  probably 
be  larger  in  this  wheel,  since  y^  is  larger ;  but  it  is  better  to  make 
the  outer  depth  a  little  too  large  than  too  small,  for  if  too  small 
eddies  might  be  formed  at  the  initial  end  of  the  bucket.  Meas- 
uring on  the  drawing,  we  find 

rp  =  0.2847  ft.  ; 
.  • .  29.82  X  18  X  0.85  X  0.284  ya  =  48  ; 

.  • .  ?/a  =  0.378  ft., 

which  value  we  use  for  the  outer  depth  between  the  crowns. 


HYDRAULIC    MOTORS.  125 

81.  To  find  the  form  of  the  croivns. — Let  the  arc  A  u,  Fig. 
53,  be  divided  into  (say)  five  equal  parts,  measured  by  trial, 
A  E  —  B  C  —  CD,  etc.  Describe  arcs  through  B  and  (7,  etc., 
with  the  centre  of  the  wheel  as  centre,  and  let 

Radius  at     A  be  ?•„ 
"       "      B,      p', 

"       <7,       p",  etc. 
Velocity  at  A,      v^ 

"         "   B,      vf,  etc. 
Angle  between  the  arcs  at  A,  y^ 

"  "        "      "     "  B,  y',  etc. 

Width  of  bucket  at  A  =  a,  —  a'  a". 
«       «         «       «  B  =  a   =V  BV, 
"       "         "       "    C  —  a"  =  c'l  c",  etc. 
Depth  between  the  crowns  at  A,yv 
"  «         «         "        "  B,  y',  etc. 

Then  for  N  buckets 

N  .  al  y^  .  vl  sin  yl  —  Q. 
N  a'  y'.  v'    sin  y'  =  Q. 

,        a.  v,  sin  y.  /  N 

.  • .  y    —  — — l- ^-7  y..  (c) 

a'  v'  sin  /  * 

But  c^i  —   c     very  nearly,  and  if  the  thickness  of  the  bucket 
r,         p 

be  neglected,  it  will  be  exact — consider  it  exact ;  then 


or, 

, ri\  vn  sin  7/a 


We  will  seek  to  produce  a  practically  uniformly  increasing 
velocity  from  A  to  u.     Then  the  increase  from  A  to  B  will  be 

.A  B 


126 


HYDRAULIC    MOTORS. 


and  the  velocity  at  B  will  be 

v'  =  v,  -j-  (i 
And  if  A  u  =  5  A  B,  then 


>)  A  B 
A  u' 


Similarly  at  (7, 


v>  —  ±Vi  +  $v9  =  14.36. 

v"  =  |  i\  +  |  v,  -  18.23. 
TABLE   XXVIII. 


By  Measurement  on  the  Drawing  . 

From  Computation 
as  above. 

From  Equation  (141). 

7i 

=  90°, 

>'i 



20    inches 

«!    =  10.50  feet 

y\ 



0.466  feet 

r' 

=  63°, 

P 

— 

23 

v'     =  14.37 

y1 

—  • 

0.332 

y" 

=  44°, 

P" 

— 

25.6 

v"    =  18.23 

y" 

rr 

0.304 

r'" 

=  30°, 

P" 

— 

27.6 

v"'   •—  22.10 

y'" 

— 

0.331 

y"" 

=  20°, 

P"" 

— 

29.0 

v""  -  25.96 

v"" 

— 

0.408 

Y* 

=  14°, 

rt 

— 

30.0 

v2    =  29.82 

y* 

— 

0.511 

The  values  of  y,,  y' ',  y'1 ',  etc.,  will  be  the  depths  of  the  buckets 
at  A,B,  C,  etc.  Since  these  will  be  the  depths  at  all  points  in  the 
circumference  of  the  arcs  passing  through  those  points  respec- 
tively, draw  a  radius  GH  and  prolong  the  arcs  to  an  intersection 
with  this  radius.  On  a  line  A  u\  Fig.  54,  equal  to  the  width  of 
the  crowns,  lay  off  the  divisions  of  the  radius,  and  through  those 
divisions  erect  ordinates  yl  =  0.-466,  y1  —  0.332,  y"  —  0.304,  etc.r 
and  trace  a  smooth  curve  through  their  extremities — these  will 
be  the  form  of  the  crowns  for  an  indefinitely  narrow  bucket. 

But  for  a  bucket  of  finite  widtli  as  we  have  in  practice,  the- 
normal  widths  should  be  measured.  These  widths  will  be  strictly 
a  curve  passing  through  the  points  A,B,  (7,  etc.,  cutting  normally 
the  traces  of  an  indefinite  number  of  buckets  between  GF  and 
a"  d,  as  shown  by  the  dotted  line  through  D,  but  it  will  be 
sufficiently  exact  to  consider  the  lines  as  straight.  "With  dividers- 
find  by  trial  the  shortest  distance  B  V  and  B  l>",  and  similarly 
for  all  the  points  C,  D,  etc.  We  find 


HYDRAULIC    MOTORS. 

TABLE   XXIX. 


127 


Distances. 

Velocities  as  found  above. 

Hence. 

a'  A  a" 

-  0.556 

0 

=  10.50 

yt    =  0.466 

V  Bb" 

=  0.572 

v' 

=  14.37 

y'     =  0.331 

c  C  c" 

=  0.492 

v" 

=  18.23 

y"    =  0.304 

d  D  d" 

=  0.396 

v'" 

=  22.10 

y'"   =  0.311 

e'  Ee" 

=  0.328 

v"" 

=  25  96 

y""  =  0.320 

pr 

=  0.284 

V<j 

=  29.82 

yz     =  0.322 

&L  —  0.378 

0.85 

The  values  of  y1  ',  y''  ',  etc.,  are  found  from  the  equations 
a'  Aa".-'o  =  V  £b.'.v'=  c'  C  c"  •<»"  =  etc.  = 


The  crowns  are  constructed  with  these  values  in  the  same 
manner  as  above,  and  are  shown  in  Fig.  5tta. 

It  will  be  seen  that  the  depths  from  the  initial  end  to  more 
than  half  the  length  are  nearly  the  same  for  a  finite  stream  as 
for  an  infinitesimal  one,  but  that  beyond  the  middle  the  depths 
are  less  for  the  latter,  and  this  corresponds  more  nearly  to  what 
they  should  be  in  practice. 

If  the  crowns  are  plane  and  parallel  yl  =  y'  =  y"  ,  etc.,  and 
the  normal  widths  will  be  inversely  as  the  velocities  along  the 
bucket  ;  hence 


B  5   =  a!  A  a"        = 


a'  A  a"  =  0.730  a'  A  a"  : 

14.37 


c'  0^=0.575  a!  A  a"  ;  d'D  d2=0.475  a'  A  a"  ;  e'  ,EX=0.404,  etc. 
In  laying  these  off  in  Fig.  55,  the  centre  line  ABC  has  been 
retained,  and  also  the  normals  B  l>  ',  C  c',  etc.,  and  with  the  ex- 
cess B  &2  =  V  B  y  —  B  ~b',  an  arc  is  described  with  B  as  a 
centre,  and  similarly  at  6Y,  D,  E,  etc.,  and  a  curve  a"  &2  c2  d* 
traced  tangent  to  the  arcs.  This  forms,  substantially,  a  back 
vane,  the  form  of  which  will  depend  upon  the  law  governing 
the  velocity  of  the  water  along  the  bucket. 


128  HYDRAULIC    MOTORS. 

Fig.  55#  shows  the  form  of  back  vane  when  the  velocity  in 
the  bucket  is  uniform  from  a"  to  a?,  and  equal  to  10.50  feet ;  and 
from  x  to  exit  increasing  to  29.82  feet. 

82.  The  pressure  at  the  entrance  into  the  wheel  will  be  given 
by  equation  (143),  making  p  —.  i\,  k  =  #,,  then 

Pi  —  pA  _|_  <y^i  _|_  £  [_  i  _  Mi  _f_  i  ^j  ^L  .   -v^Lr-0  =  2894. 

The  pressure  at  exit  will  be 

P*  =P*+  $  kv 

If  the  wheel  be  submerged  one  foot,  then 
p^  =  2116  +  62.4  X  1  =  2178.4  pounds  per  square  foot. 

83.  To  find  the  path  of  the  water  in  reference  to  the  earth. 
The  path  is  determined  on  the  supposition  that  the  velocity  of 
the  water  is  uniformly  increasing  as  it  passes  through  the  wheel, 
as  given  in  Table  XXIX.,  on  page  127,  but  is  uniform  in  pass- 
ing from  A  to  B,  B  to  (7,  etc.,  and  then  proceeding  as  in  Article 
50.     The  result  is  shown  by  the  line  J  K,  Fig.  53. 

84.  The  diameter  of    the  shaft  for   ample   security  may  be 
found  from  the  equation 


V 
d  =  V 


100  HP    =  4  =  4  inches 

1.04 


nearly,  and  3|  inches  will  be  safe. 

The  wheel  must  be  so  secured  to  the  shaft  that  it  will  not  get 
free  nor  even  slip.  It  may  be  secured  by  a  shrunken  band,  as 
in  Fig.  19,  or  by  a  key,  as  in  Fig.  31.  A  key  seat  requires  the 
cutting  away  of  some  of  the  material  of  the  shaft,  and  at  this 
point  it  may  be  advisable  to  make  it  4  inches,  even  if  it  be  less 
for  the  remaining  part. 


HYDRAULIC    MOTORS.  129 

85.  Turbines  are  now  frequently  mounted  on  a  horizontal  shaft, 
producing  practically  the  same  efficiency  as  if  vertical,  and  fre- 
quently are  in  pairs,  the  wheels  facing  each  other,  so  as  to  relieve 
the  shaft  bearings  of  end  or  axial  pressures. 

The  evolution  of  the  modern  American  wheel  seems  to 
have  begun  with  the  "  Swain"  wheel,  which  was  being  devel- 
oped from  about  1862  to  1875.  This  in-and-axial  flow  wheel  in 
its  final  form  had  a  high  efficiency,  as  shown  in  the  test  already 
given,  and  its  construction  was  comparatively  cheap.  The  wheel 
was  cast  solid,  with  less  and  deeper  buckets  than  the  Fourneyron 
and  wider  openings  for  discharge,  and  produced  at  a  cost  of,  say, 
J  to  -J-  of  that  of  the  built-up  wheels  of  Boyden  and  Francis,  and 
yielding  a  higher  efficiency.  Then  followed  the  Risdou,  with  its 
reputed  87  per  cent,  efficiency,  then  the  Hercules  (1876)  with  a 
still  smaller  diameter  and  still  deeper  buckets  and  large  discharge,, 
cast  solid,  and  yielding  a  "  test "  efficiency  of  87  per  cent. ;  and 
this  was  quickly  followed  by  the  "  Victor"  on  the  same  general 
plan,  but  with  different  buckets  and  gate,  and  yielding  as  high 
if  not  higher  efficiency.  Other  wheels  of  good  repute  have 
been  produced  during  this  evolution.  The  Leffel  wheel,  in  which 
the  inward  and  downward  passages  were  separated  by  a  dia- 
phragm, and  the  "  Humphrey"  wheel,  herein  described,  and  still 
others,  in  all  of  which  more  than  80  per  cent,  efficiency  has  been 
delivered,  so  that  the  selection  of  a  good  wheel  by  purchasers 
must  be  made  on  other  grounds.  The  capacity  of  a  wheel  in- 
producing  a  given  power  is  one  of  these  elements. 

The  following  is  an  approximate  comparison  of  the  wheels  as 
they  have  usually,  or,  perhaps,  it  is  safer  to  say  as  they  were 
formerly,  made,  for  the  proportions  of  any  one  of  them  may  be 
changed,  and  before  this  time  may  have  been  so  changed  as  to 
produce  different  figures.  The  comparison  is  for  a  fall  of  24  feet 
and  diameter  of  wheel  of  about  36  inches. 


130 


HYDRAULIC    MOTORS. 

TABLE  XXX. 


Cu.  ft.  water 
per  sec. 

H.P. 

Bovden-Fourneyron      

30 

65 

Risdon              

50 

115 

Swain        

100 

220 

Hercules       

108 

235 

Leffel-Samson    

110 

240 

Victor  

111 

241 

According  to  these  figures,  the  purchaser  must  choose  his  wheel 
upon  other  considerations  than  mechanical  efficiency  or  economy 
of  space. 

86.  Niagara  "Wheel. — The  Niagara  Falls  Power  Co.  -propose  to 
utilize  some  100,000  horse-powers  of  the  Falls  of  Niagara.  For 
this  purpose  a  short  canal  or  bayou  250  feet  wide  and  12  feet 
deep  a  mile  or  more  aboye  the  crest  of  the  falls,  excavated  for 
this  purpose^  conducts  the  water  from  Niagara  River  to  vertical 
shafts,  in  the  lower  ends  of  which  are  placed,  or  are  to  be  placed, 
turbines,  which  will  have  a  head  of  about  136  feet.  The  tail 
races  conduct  the  escaping  water  to  a  tunnel  7000  feet  long,  21 
leet  high  and  about  18  feet  wide,  discharging  into  the  river  a 
short  distance  below  the  falls. 

In  this  country  most  turbines  are  made  from  fixed  patterns  of 
definite  sizes  to  suit  average  conditions,  are  turned  out  in  quantities, 
and  kept  in  stock,  like  store  goods,  to  be  purchased  when  wanted ; 
but  in  Europe  they  are  more  generally  made  to  order,  designed 
for  the  particular  place  and  conditions.  So  when  the  Niagara 
commission  called  for  plans  of  wheels,  the  American  manufac- 
turers submitted  their  trade  catalogues,  and  the  European  manu- 
facturers submitted  special  designs.  Among  the  designs  con- 
sidered was  the  American  twin  turbine  on  a  horizontal  shaft, 


HYDRAULIC    MOTORS.  131 

with  belt  or  rope  transmission,  but  the  difficulties  encountered 
were  so  great  that  such  plans  were  finally  abandoned,  and  the 
design  of  Messrs.  Faesch  &  Piccard.  of  Geneva,  Switzerland, 
was  accepted  ;  the  plan  of  which  is  shown  in  Fig.  56,  for  which 
the  author  is  indebted  to  the  courtesy  of  Colernan  Sellers,  E.D., 
Prof'r  of  Practical  Eng'g  Stevens  Institute  and  Pres.  and  Chief 
Eng'r  of  The  Niagara  Falls  Power  Co.  The  water  passes  from 
the  lower  end  of  the  penstock  A,  Fig.  57,  into  the  casing  B  B, 
thence  through  the  distributers  b  b  5,  three  of  which  are  at  the 
upper  end  of  the  case  and  three  below ;  thence  through  the 
buckets  a  a  a.  The  wheel  is  divided  into  six  elementary 
wheels  by  transverse  discs,  three  of  which  are  above  where  the 
water  enters  and  three  below.  The  gate  c  c  is  a  cylinder  ex- 
terior to  the  wheel,  and  opens  the  wheel  passages  by  being 
moved  downward.  The  shaft  C  passes  through  the  central 
part  of  the  case,  to  the  former  of  which  the,  crowns  of  the 
wheel  are  firmly  secured.  The  upper  crown  is  solid,  but  through 
the  upper  end  of  the  case  are  openings  d  c7,  through  which 
water  may  pass  into  the  space  between  the  upper  end  of  the 
case  and  the  upper  crown  of  the  wheel,  which,  acting  by  upward 
pressure,  supports  a  part,  or  all,  of  the  weight  of  the  wheel, 
shaft,  and  attachments.  The  lower  end  of  the  case  is  solid, 
but  the  lower  crown  has  openings  h  A,  so  that  any  water  enter- 
ing the  space  above  it  may  readily  escape  so  as  not  to  produce 
a  downward  pressure  upon  the  wheel.  The  lower  end  of  the 
case  is  supported  by  three  rods  g  g  extending  through  the 
case,  two  of  which  are  shown  in  the  figure.  The  gate  is  regu- 
lated by  a  delicate  and  efficient  regulator,  so  that  the  deviation 
from  the  velocity  desired  is  less  than  i  per  cent,  when  the  load  is 
increased  or  decreased  by  25  per  cent. 

The  figures  in  Fig.  56  were  used  in  the  construction  of  the 
wheel.  The  wheel  was  made  of  bronze,  and  the  buckets,  parti- 
tions, and  crowns  immediately  above  and  below  the  buckets  are. 
solid  in  one  casting. 


132  HYDKAULIC    MOTORS. 

The  terminal  angle  of  the  guides  is  given  as  19°  6',  and  of  the 
buckets  13°  17^',  both  of  which,  being  comparatively  small,  are 
favorable  to  economy,  and  the  comparatively  small  initial  angle 
of  the  buckets,  69°  20'  (or  less),  makes  it  a  wheel  of  consider- 
able pressure. 

The  principal  data  are  : 

External  diameter, 2/'2  =  6  ft.  3    in. 

Internal         "  2^  =  5  "  3     " 

Width  of  crown, 6     " 

External  diameter  of  distributing  chamber,         5  "  2-J  " 
Internal         "  4  "  4     " 

Clearance  of  wheel, Ty  " 

"Width  of  distributing  chamber, .     .     .  ^iVu 

Diameter  of  penstock, 7  "  6     " 

Number  of  buckets,       ......  N~    =  32. 

Number  of  guides, N1  =  36. 

Depth  of  each  chamber  a,       ....  3f  " 

Clear  depth  of  six  chambers  a  a  a,  .     .  1.81  ft. 

Thickness  of  the  horizontal  partitions,  each,  1  " 

Terminal  angle  of  guides,       .     .     .     .  a  =  19°  6' 

Initial  angle  of  buckets,  about,   .     .     .  ;',  =  69°  20' 

Terminal  angle  of  buckets,     .     .     .     .  )/2  =  13°  17J' 

Total  head,  about, //  =  136  ft. 

In  regard  to  the  head  some  writers  have  given  it  as  136  feet, 
while  others  have  called  it  140  feet.  It  is  not  necessarily  con- 
stant, varying  somewhat  with  the  height  of  water  in  the  river. 
Clemens  Herschel,  one  of  the  engineers  of  the  company,  in 
Gassier'' s  Magazine,  of  July,  1895,  says  :  "The  wheels  will  dis- 
charge 430  cubic  feet  per  second,  and  acting  under  136  feet  head 
from  the  surface  to  the  centre  of  the  wheel,  will  make  250  rev- 
olutions per  minute  ;  at  75  per  cent,  efficiency  will  give  5,000 
horse-power."  Mr.  Stetson,  an  officer  of  the  company,  in  the 
same  magazine,  speaks  of  140  feet.  If  the  wheel  discharges  430 


HYDRAULIC    MOTORS.  133 

cubic  feet,  the  velocity  in  the  penstock  will  be  between  9  and  10 
feet  per  second,  which  will  be  equivalent  to  more  than  one  foot 
head,  so  that  the  effective  head  will  be  over  137  instead  of  136. 
The  head  at  the  lower  end  of  the  wheel  will  be  about  10  feet 
greater  than  that  at  its  upper  end,  and  the  mean  head  will  be  a 
little  lower  than  to  centre  of  the  wheel.  We  will  use  for  com- 
putation 

H  =  138  feet. 

The  initial  angle  of  the  buckets  is  marked  as  110°  40',  the 
supplement  of  which  in  our  notation  is  yl  =  69°  20'.  But  this 
is  the  mean  angle  between  the  face  and  back  of  the  vane.  We 
made  a  computation  for  the  efficiency  and  speed  using  this  angle ; 
but  it  made  the  efficiency  very  high,  85  per  cent.,  and  the  speed 
too  low,  232  revolutions  per  minute.  With  another  computation 
with  large  assumed  friction  of  the  water,  we  found 

E  =  79  per  cent,  hydraulic  efficiency, 
and 

n  =  228  revolutions  per  minute. 

These  revolutions  are  much  less  than  those  given  in  Fig.  56,  or 
than  has  been  found  in  practice.  The  face  angle  of  the  bucket, 
measured  on  a  drawing  of  the  wheel,  is 

X,  =  51°, 

and  with  this  angle  we  found  results  agreeing  fairly  with  those 
found  by  Dr.  Sellers,  as  communicated  to  the  author,  so  we  have 
used  this  value.  This  is  our  first  opportunity  of  determining 
from  actual  trials  whether  yl  for  a  finite  stream  should  be  the 
angle  made  by  the  face  of  the  bucket  with  a  tangent  to  the 
wheel,  or  the  mean  angle,  and  the  indication  is  quite  clear  that  it 
should  be  the  angle  of  the  face.  With  the  data 

r,  =  2.625  ft.         a  =  19°         /*,  =  0.10 
r,  =  3.125  "         YI  =  51°         ;i,  =  0.15 
H=  138      "        >,  =  13°  IT   g  =  32.16  ft., 
2  r  =  38  in.  outside  diameter  of  the  tubular  part  plihe  shaft. 


134  HYDRAULIC    MOTORS. 

t  =  f  in.  thickness  of  tube  of  shaft. 

d  =  11  in.  outside  diameter  of  solid  part  of  shaft. 

"We  find  from  equations  (15)  to  (22)  ;  and  the  equation  preced- 
ing (144) : 

M *  =  —  0.478       N*  =  -  0.156  tf  =  642.99. 

.  • .  GO'  =  25.357  angular  velocity  per  second ; 
.  • .  N  =  242.2  rev.  per  minute  ; 

E  =  0.8108,  or  81  per  cent,  efficiency; 

v1  =  22.98  ft.  velocity  of  entrance  into  bucket ; 

v^  —  82.91   "  terminal  velocity  in  the  bucket ; 

V  =  55.27  "  velocity  of  quitting  the  guide; 

T^  —  19.13   «  quitting  velocity  in  reference  to  the  earth  ; 

8  =  85°  35'; 
H.P.  —  5,500  horse-power ; 

Q  =  433.27  cu.  ft.  per  second  ; 

y,  =  1.55  ft. 

y,  =  1.52  « 

oor  rl  =  66.56  ft.  velocity  of  inner  rim ; 
to'  7\  =  79.24  "  "   outer    " 

J  —  867  Ibs.  stress  on  the  tubular  part  of  the  shaft. 

Several  of  these  results  differ  perceptibly  from  those  given  in 
Fig.  56.     Assuming  250  revolutions  per  minute,  we  have 

250  X  2  n  ,  -, 

GO  =  CL =  26.16  ft.  per  second. 

60 

GO  r,  =  68.30  ft.  velocity  of  initial  rim. 
GO  7\  =  81.81  "         "         "    terminal  rim. 

And  if  y,  =  180°  -  110°  40'  =  69°  20'  and  a  =  19°  00r,  the 
triangle  of  velocities  gives 

V  =  64.31  ft.  the  velocity  of  quitting  the  guides. 

vl  —  22.5  ft.,  nearly,  the  initial  velocity  in  the  bucket. 


HYDRAULIC    MOTORS.  135 

And  if  the  ratio  of  the  initial  normal  section  of  the  bucket 
to  that  of  the  terminal  be  as  4.275  to  1.25  —  3.42,  then 

^2  =  3.42  X  22J  =  76.95  ft.  terminal  velocity  in  the  bucket, 
V^  =  19  ft.  actual  velocity  of  quitting, 
and 

8  =  113°,  direction  of  F2. 

These  figures  agree  so  nearly,  almost  exactly,  with  those  given 
in  Fig.  56,  that  we  assume  that  they  were  obtained  in  this  man- 
ner. If  the  wheel  makes  250  revolutions  per  minute  when  pro- 
ducing its  highest  efficiency,  under  a  head  of  136  feet,  and  the 
other  data  be  as  given  or  assumed,  the  solution  is  correct  ;  but 
otherwise,  it  is  only  an  approximation,  more  .or  less  rough. 
As  stated  above,  the  data  in  Fig.  56  gives  about  232  revolutions 
per  minute  for  best  effect,  and  our  computation  with  y^  —  51° 
gives  242  ;  so  that  if  it  had  75  per  cent,  efficiency  and  gave  5,000 
horse  -power  at  250  revolutions,  it  ought  to  have  a  higher  effi- 
ciency and  greater  capacity  at  a  slower  speed.  If  430  cubic  feet 
were  discharged  at  a  velocity  of  76.95  feet,  through  32  buckets 
each  1J  inches  wide,  the  depth  should  be 

° 


A  of  1.25  X  32  X  76.95 

The  actual  depth  of  the  six  chambers  is  1.81  feet  ;  hence  the 
capacity  of  the  wheel  should  exceed  5,000  horse-power  in  the 
ratio  1.81/1.68,  giving  5320  horse-power. 

This  assumes  that  the  buckets  are  properly  made.  It  appears 
that  the  cross-section  a  d,  Fig.  56,  is  slightly  less  than  that  at 
f  e,  whereas  the  former  ought  to  be  perceptibly  larger,  since  the 
velocity  of  the  water  increases  as  it  goes  outward,  so  that  if  the 
section  at  a  d  is  filled,  that  at/  e  will  not  be  full,  and  the  wheel 
will  be  a  "  pressure"  wheel  from  the  initial  element  to  a  d,  and 
one  of  "  free  deviation"  from  a  d  to  exit.  This  being  the  case, 


136  HYDKAULIC    MOTORS. 

as  determined  from  the  plan  of  the  wheel,  the  correct  depth 
would  be  found  by  using  the  velocity  at  a,  which  will  be  some- 
what less  than  at  e,  and  also  the  width  at  «,  which  is  also  some- 
what less  than  at  e /  and  both  these  elements  conspire  to  make 
the  depth  y^  greater  than  1.68  as  computed.  We  would  modify 
the  design  of  the  wheel  by  terminating  the  long  arc  at,  or 
near,/*,  and  using  a  shorter  radius  to,  or  near,  c.  Figs.  54  and  54& 
are  suggestive  of  a  better  form  than  that  given  in  this  wheel. 
But  retaining  the  width  e  f  and  increasing  it  at  a  d  ought  to  in- 
crease the  capacity  of  wheel  somewhat  without  decreasing  the 
efficiency. 

We  now  return  to  our  computation.  If  the  results  obtained 
from  equations  (15)  to  (21)  do  not  agree  with  those  found  in  the 
wheel,  then  equations  (56)  to  (62)  must  be  used,  since  the  sections 
of  the  buckets  are  fixed.  First,  the  cross- sections  of  the  initial 
and  terminal  sections  of  the  buckets  must  be  inversely  as  the 
velocities  vjv,  =  82.91/22.98  =  3.679. 

To  determine  the  ratio  of  the  sections  of  the  stream  we  assume 
that  the  initial  section  of  the  stream  is  the  same  as  that  of  the 
bucket,  but,  as  has  been  shown  above,  the  terminal  section  of 
the  stream  is  less  than  that  of  the  bucket ;  and  to  find  what  the 
former  is — or  what  the  section  of  the  bucket  ought  to  be — re- 
quires a  knowledge  of  the  capacity  of  the  wheel.  This  requires 
an  extended  analysis,  not  here  given,  according  to  which  and  to 
other  information,  it  is  for  138  foot  head  about  5,500  horse- 
powers ;  and  we  wdll  assume  this  value  for  the  present  computa- 
tion, and  test  its  correctness  by  the  results  which  follow  : 

The  volume  of  water  flowing  through  the  wheel,  in  produc- 
ing 5,500  horse-power,  will  be 

ft.  lb. 
per  sec. 
H.P.  per  H.P. 

Q  =      5'500  X  55°       100  =  433  cu.  ft.  per  sec. 
81  X  138  X  62.4 

per  ft.  \vt. 

cent.  cu.  ft. 


- 


PLAN  OF  WHEEL 

OF  THE 

NIAGARA    POWER    CO. 

/        Estimated  5000 


Fra.  56. 


HYDRAULIC    MOTORS.  137 

The  terminal  velocity  being  83  ft.  to  the  nearest  entire  foot, 
as  found  above,  the  aggregate  depth  of  the  six  chambers  being 
1.81  feet,  the  aggregate  width  of  the  32  buckets  should  be 

433  +  (83  X  1.81)  =  2.882  ft., 
and  the  width  of  each  would  be 

12  X  2.882  -7-  32  =  1.081  in., 

instead  of  1J  in.  as  marked  on  the  plan. 

The  thickness  of  the  initial  end  of  the  partitions  between 
the  buckets  as  measured  on  the  drawing  is  -J-  of  an  inch  each,  and 

for  the  32  buckets  the  aggregate  thickness  will  be  32  X  --  ••  •  ft. 

LA  /\  O 

If,  therefore,  the  buckets  were  indefinitely  narrow,  the  aggregate 
thickness  of  all  the  partitions  being  the  same,  the  aggregate  nor- 
mal widths  would  be 

2  n  r,  sin  51°  —  32  .  .  _J     -  =  12.27  ft, 
12x5 

and  the  ratio  of  the  sections  in  this  wheel  being  as  their  widths, 
we  have 

£,       .   12.27  _  .  ,  95 

'  - 


But  for  buckets  of  finite  width,  the  ratio  is  found  more  accu- 
rately by  tracing  a  curved  line  cutting  normally  the  traces  of  the 
buckets,  if  there  were  an  indefinitely  large  number.  This  proc- 
ess gives  a  ratio  of  about  4  or  a  little  more.  We  find  that  the 
results  do  not  differ  largely  for  the  ratios  4  and  4.25,  except  for 
the  value  of  v^  the  initial  velocity  in  the  bucket  ;  so  we  take 
the  ratio  4  as  representing  more  accurately  the  actual  wheel, 
and  as  being  sufficiently  accurate  for  our  purpose,  and  this  dif- 
fers so  much  from  the  inverse  ratio  of  the  velocities  (3.679)  as  to 
make  a  computation  with  equations  (56)  to  (62)  desirable.  With 
the  data 


138 


HYDRAULIC    MOTORS. 


K  =  kj\  =  4.00,        ^  =  0.10, 
r,  =  2.675,  ^  =  0.10,  Yl  =  51°, 

r,  =  3.125,  J7  =  138  ft.         r,  =  13°  IT, 

we  find 

c  =  1/JT=  0.25,          a  =  1.11203,       £  =  3.092, 
J.  =  0.503,  B  =  0.897,         J*  =  —  4.221. 

<72  =  -  3.9678  F*  =  1.602,          £  =  2.531. 

.  • .  ^max  =  0.834, 
a?  =  26.45, 
n  =  252.5, 
F  =  54.54, 
v,  =  85.29  ft., 
vl  =  21.32  " 
F2  =  19.62  " 
a/  r,  =  69.43  " 
GO'  r,  =  82.65  « 
(9  =  88°  53r, 
a  =  14°  10r, 
H.  P.  =  5,500, 

Q  =  433  cu.  ft.  per  sec., 
Least  breadth  of  bucket  =  1.04  in. 

No  allowance  is  here  made  for  leakage  through  the  TVm. 
clearance  of  the  wheel.  There  are  three  such  clearances  for  the 
escape  of  water,  two  at  the  lower  part  and  one  at  the  upper  part 
of  the  wheel.  From  equation  (44)  it  is  found  that  the  pressure 
at  entrance  into  the  wheel  is 

pl  =  7,468  pounds  per  sq.  foot ; 

hence,  if  the  coefficient  of  discharge  be  0.80,  the  volume  of  dis- 
charge will  be 


FIG.  57. 


HYDRAULIC    MOTORS  139 


/  - 

y  ^gp-i  =  15.4  cu.  ft.  per  sec. 


Hence,  the  quantity  of  water  passing  into  the  penstock,  when 
running  with  full  gate  and  at  best  effect,  should  be  about 

Q  =  433  +  16  =  449  cu.  ft.  per  sec. 
This  computed  leakage  will  be 

TO  -  °-0344'  ' 

or  3.44  per  cent,  of  the  water  delivered  to  the  penstock  ;  or  96.56 
per  cent,  of  the  water  delivered  to  the  penstock  passes  through 
the  wheel  ;  hence  the  efficiency  of  the  wheel  system  referred  to 
the  water  consumed  will  be 

E  '=  83.4  X  .9656  =  80.53  per  cent. 

The  terminal  angle  of  the  guide  (19°)  is  somewhat  larger  than 
given  by  theory. 

We  make  the  following  abstract  of  a  statement  in  regard  to  a 
test  furnished  by  Dr.  Sellers. 

At  the  time  of  the  test  the  total  head  from  the  surface  of  the 
water  above  the  penstock  to  the  centre  of  the  wheel  was 

II  —  135.113  ft., 
and  the  water  delivered  to  the  penstock  per  minute  was 

Q  =  26867  cu.  ft.  per  min.  =  447.8  cu.  ft.  per  sec., 
and  the  theoretical  horse-power  of  the  water, 

HP—  447.8  X  135.113  X  62.3  =  ^^ 
550 

There  was  an  electrical  output  of  5335  horse-powers  ;  hence 
the  actual  efficiency  of  the  wheel  and  dynamo  combined  was 

77V 

= 


140  HYDKAULIC    MOTORS. 

or,  77.85  per  cent.  ;  and  if  the  dynamo  yielded  97  per  cent,  as 
guaranteed  by  the  makers,  then  the  efficiency  of  the  wheel  sys- 
tem, including  friction  and  leakage,  would  be 

E  =  TT-85  =  80.26  per  cent, 
0.97 

and  the  power  delivered  at  the  upper  end  of  the  shaft  would  be 
H.  P.  =  ^  =  5,500. 


The  head  during  this  test  was  less  than  that  assumed  in  the 
computation,  but  if  1.4  ft.  for  the  head  due  to  the  velocity  in 
the  penstock,  be  added,  the  effective  head  will  be  136.5  ft., 
which  is  only  1.5  ft.  less  than  the  effective  head  assumed.  This 
difference  will  not  affect  the  efficiency,  but  would  affect  the  com- 
parative speed.  The  speed  was  not  measured,  but  was  regulated 
for  about  245  to  250  revolutions  per  minute,  and  the  experi- 
mental efficiency  and  power  involve  an  assumption ;  and  the  theo- 
retical computation  is  founded  on  the  supposition  that  the  wheel 
is  a  pressure  wheel  throughout ;  so  that  it  cannot  be  said  whether 
a  more  exact  analysis  would  agree  more  nearly  with  a  test  ex- 
periment, if  the  data  were  precisely  the  same  and  the  quantities 
directly  measured.  As  they  stand,  the  two  results — theory  and 
experiment — agree  remarkably  well.  The  indication  is — the  re- 
sistances are  less  than  those  assumed,  the  leakage  greater  than 
that  computed,  and  the  hydraulic  efficiency  greater  than  85  per 
cent,  of  the  power  of  the  water  passing  through  the  wheel. 

The  volume  of  water  may  now  be  recomputed,  and  will  be 

O  =          5,500  X  550          _ 
V  "     62.4  X  138  X  0.8053 

which  is  3  cu.  ft.  more  than  that  before  found,  which  differ- 
ence  is   chiefly   due   to   the   difference   in   the   efficiency  used 


FIG,  58. 


HYDRAULIC    MOTORS.  141 

in  the  computation.     Using  this  result,  the  terminal  normal  width 
of  the  bucket  will  be  : 


Width  =  _        x  12  _  =  1.04  in., 
1.81  X  85.29  X  32 

which  is  0.04  of  an  inch  less  than  that  found  by  the  former 
computation,  which  is  due  to  the  larger  terminal  velocity  now 
found.  This  again  emphasizes  the  remark  previously  made  in 
regard  to  increasing  the  capacity  of  the  wheel. 

According  to  our  computation,  the  velocity  of  the  water  in  the 
penstock  will  be 


The  velocity  as  it  enters  the  case  will  be  .....  12.2  " 
"  "  in  the  case  just  before  entering  the  dis- 

tributers will  be     ........  28.5  " 

"  "  entering  the  distributers  will  be  .  ,  .  .  30.4  " 

"  "  quitting  "  ...  -  55.3  " 

"  "  entering  the  wheel  relative  to  the  bucket 

will  be      .......  ......  21.3  " 

"         "        quitting  the  bucket  will  be     .     ..    .     .     .  85.3  " 

"  "  u  "  wheel  in  reference  to  the 

earth  will  be      .........  19.6  " 

The  main  part  of  the  shaft  is  a  tube  of  steel  rolled  and  without 
longitudinal  riveted  seam,  38  in.  outside  diameter  and  j  in. 
thick.  There  are  two  solid  parts  joining  the  tubular  parts,  as 
shown  in  Fig.  59,  which  form  journals  for  the  support  of  the 
shaft  and  wheel,  and  are  11  in.  in  diameter,  one  of  which  is 
shown  in  Fig.  58.  The  moment  of  stress  is  given  in  Article  49, 
and  is 

12Pa  =  63'000  H'P'  inch  pounds. 
n 

For  the  resistance,  let 
r  be  the  mean  radius  of  the  tubular  shaft  in  inches  ; 


142  HYDRAULIC    MOTORS. 

t,   the  thickness  of  the  tube  ; 

t/,  the  modulus  of  torsional  shear  ; 

then  for  a  thin  tube 

2  7t  r  .  t 
will  be  the  sectional  area  of  the  tube, 

2  n  T  t  .  J 
will  be  the  resisting  force  of  the  tube,  and 

2  TT  r  t  J.r 
will  be  the  moment  of  resistance  ; 

,  .   7       63,000/7:  P. 

.  •  .  2  TT  r  t  J  =  -  - 


r  _  31,500  II.  P. 
* 


which  in  this  case  becomes, 

J=       .  81600  X5,500_.      =  867poimd, 
3.U  X  ISf  X  f  X  252 

The  torsional  stress  on  the  solid  part  will  be  given  by  the  equa- 
tion 

63,000  H.P.  =jxjjp 
n 

in  which  R  is  the  external  radius  of  the  solid  part  and  is  5£  in.  ; 

T        ,  126,000  H.P. 

,  •  .  J  =  8  -  -  -  =  5,260  pounds. 
it  n  .  11s 

The  resistance  to  shear  of  steel  or  iron  in  large  masses  is  not 
well  known.  If  homogeneous,  theory  indicates  that  it  will  be 
£  of  the  tenacity  of  the  material,  and  experiments  indicate  that 
the  shearing  resistance  is  nearly  the  same  as  that  of  the  tenacity. 
The  tenacity  of  mild  steel  is  65,000  pounds  and  upward  per 
square  inch  ;  hence  its  shearing  strength  ought  to  be  50,000 
pounds  at  least,  according  to  which  the  solid  part  will  be  strained 


HYDRAULIC    MOTORS.  143 

to  about  -j-^  of  its  ultimate  strength  when  running  steadily  and 
delivering  5500  horse-powers,  which  is  no  more  than  ought  to 
be  allowed  for  safety,  considering  that  in  starting  and  stopping 
and  for  variations  of  loads,  the  stress  may  be  considerably  in- 
creased. The  stress  on  the  tubular  part  is  small  compared  with 
that  on  the  solid  part — less  than  ^  as  great.  If  the  shaft  be  a 
uniform  tube  f  in.  thick,  19  in.  radius,  140  ft.  long,  and  if  the 
modulus  of  elasticity  to  shear  be  10,000,000  pounds,  then  will 
the  amount  of  torsion,  when  running  steady  at  252  revolutions, 
delivering  5500  horse-powers,  be 

.    63,000  X  5,500  X  140  x  12    =  Q 
"  10,000,00,0  X  2   n  r  t  .  r* .  n 

which  is  the  arc  for  radius  unity. 
The  number  of  degrees  will  be 


9 

3.14 


X  180  =  4°  13'. 


Fig.  59  shows  the  penstock,  shaft,  and  relative  position  of  the 
wheel.  They  are  supported  by  heavy  cast-iron  beams  resting  on 
the  solid  rock. 

Pressure  due  to  deflecting  a  stream. 

87.  The  pressure  resulting  from  the  deflection  of  stream  of 
water  may  be  determined  as  follows  :  Let  a  particle  whose  mass 
is  in  enter  a  stationary  tube  with  a  velocity,  -y,  and  follow  the 
tube  to  its  exit.  Let  the  tube  be  frictionless,  then  will  the  veloc- 
ity be  v  in  reference  to  the  tube  from  entrance  to  exit.  Let  the 
tube  be  the  arc  of  a  circle  with  0  as  the  centre.  The  centrif- 
ugal force  will  be  radially  outward  and  equal  to 

v2 

m  —  ; 
r 

which  will  be  the  pressure  against  the  outside  of  the  tube,  and 
may  be  represented  in  magnitude  and  direction  by  the  line  A  B 


14-i  HYDRAULIC    MOTORS. 

on  the  radius  0  A  prolonged.  The  centripetal  force  will  be  of 
equal  magnitude,  and  will  be  the  reaction  of  the  tube  acting 
upon  the  particle  toward  the  centre  0.  Assume  that  the  particle 
fills  the  tube  for  a  distance  d  s  ;  then  if 

k  be  the  cross-section  of  the  tube  ; 

d  the  weight  of  unity  of  volume  ; 

6  the  angle  D  0  A,  Fig.  60  ; 

r  the  radius  0  A  ; 

<p  the  centrifugal  force  ;  then 
d  s  =  r  d  0 


gr 

Resolve  this  force  into  two  forces,  B  C  parallel  to  0  x  and 
A  C  perpendicular  thereto,  then  will  the  sum  of  all  the  pressures 
parallel  to  0  x  due  to  the  particle  in  passing  from  D  to  A  be 


=  Sky*    C 

g   Jo 

(I  _  cos  0), 


sin  6  d  6, 

«y  u 

dkv* 


g 

where  X&  represents  the  a?-  component  of  the  pressure. 

Instead  of  a  single  particle  moving  in  the  tube  let  a  stream  of 
liquid  flow  through  the  tube,  and  let 

M  be  the  mass  flowing  into  the  tube  per  second,  then 

M  =  Shv. 


hence  the  constant  pressure  in  the  direction  0  x  will  be 

X9  =  M  v  (1  -  cos  6).  (147) 

If 

8  =  %*, 
then 

Xln  =  M  v,  (US) 


HYDRAULIC    MOTORS  A  U        LV_:  *45 

5£ALIFORH\^ 

or  the  resultant  pressure  equals  the  momentum^  of  the  jet  per 
second. 
If 

6    =    7t 


(149) 
or  the  resultant  pressure  will  be  twice  the  momentum  of  the  jet. 

88.  Now  assume  that  the  tube  moves  in  the  direction  0  x  with 
an  uniform  velocity  u,  and  that  M  is  the  mass  of  the  stream  per 
section  k  entering  the  tube,  the  relative  velocity  will  be 

v  —  u, 
and  the  resultant  pressure  in  that  direction  for  an  arc  subtend- 


ing 


6 


M(v  —  u)  (1  —  cos  0),  (150) 

and  if  0  be  90°  the  work  per  second  will  be 

P  u  =  M(v  —  u)  u.  (151) 

This  will  be  a  maximum  for  u  =  •§  v,  for  which 

Pu  =  ±  M  v\  (152) 

or  one  half  the  energy  of  the  stream  will  be  utilized  in  doing 
work,  the  other  half  remaining  in  the  stream.  This  result  may 
be  deduced  directly,  thus  :  The  relative  velocity  at  the  entrance 
D  being  v  —  u  =  J-  v,  and  the  tube  being  frictionless,  the  veloc- 
ity at  exit  will  also  be  J-  v  ;  hence  at  E  the  water  will  have  a 
component  of  \  v  parallel  to  0  x  and  another  of  J-  v  perpendic- 
ular to  the  same,  the  resultant  of  which  will  be 


and  the  energy  will  be  J  J/V  ;  the  remaining  half  being  trans- 
formed into  work.  If  the  mass  be  that  which  passes  a  fixed 
point,  represented  by  M '  having  the  section  &,  then  will  the 
mass  entering  the  tube  be 

Jf'(l-|),  (153) 

and  the  work  will  be 


146  HYDRAULIC    MOTORS. 

P  u  =  M'  L_    —L  u,  (154) 

srfiich  will  be  a  maximum  for  u  =  J  v  ; 

.-.  Pu  =  —  M'v*.  (155) 

27 

But  this  is  referring  the  energy  to  a  mass,  a  part  of  which  is 
not  delivered  to  the  wheel,  a  method  which  is  not  ordinarily  prac- 
tised. Moreover,  in  practice  floats  or  buckets  are  usually  made 
to  follow  each  other  so  closely  that  the  stream  of  the  section  of 
the  buckets  is  assumed  to  be  all  delivered  to  the  wheel,  and  equa- 
tion (151)  is  applicable.  This,  then,  is  the  case  of  an  impulse 
wheel,  or  wheel  of  free  deviation,  in  which  the  direction  of  the 
terminal  element  is  ;/2  =  90°.  By  assuming  that  the  water  im- 
mediately in  front  of  the  floats  in  a  "  paddle"  or  undershot 
wheel,  Fig.  61,  forms  a  guide  for  the  water  which  subsequently 
follows,  the  analysis  for  the  work  of  the  plane  float  undershot 
wheel  results  in  equation  (151),  and  hence  such  a  wheel  cannot 
utilize  more  than  one  half  the  energy  of  the  stream  ;  and  as  there 
are  inevitable  wastes  from  the  escaping  water  through  the  clear- 
ances, the  practical  efficiency  will  be  still  less,  and  experiments 
have  shown  it  to  be  about 

0.30  J/V,  (156) 

and  except  for  wheels  well  made,  mechanically,  it  will  be  still 
smaller.  We  have  thus  brought  this  wheel  within  the  analysis 
of  the  turbine  of  free  deviation. 

89.  It  has  been  stated  in  Article  6  that  equation  (16) 
gives  oo,  =  0,  for  r,  —  90°.  If  ;/2  =  0  be  substituted  in  (15) 
it  becomes 

E=ZM'.rt'a/,  (157) 

which  increases  indefinitely  with  the  square  of  the  speed  of  the  wheel, 
and  hence  has  no  maximum.  Also  if  a  =.  0  and  yl  =  1 80°,  M  and 


FIG.  59. 


ffse  LIB/ 

"*£          OF   THB 

UNIVERSITY 
& 


HYDRAULIC    MOTORS.  147 

N  contain  ambiguous  terms  of  the  form  —  ,  some  of  which  when 

evaluated  give  infinity  ;  so  that  the  general  equation  seems  to 
fail  for  these  particular  values  ;  because  there  is  no  maximum  for 
this  case.  But  for  the  wheel  of  "  free  deviation"  there  is  a 
maximum  for  this  condition  as  shown  by  the  following  analysis  : 
For  this  case  the  velocity  V  will  depend  directly  on  the  head, 
and  making  pl  =  pa,  and  A,  =  H  in  equation  (4),  we  have 

(1  +  O  V  =  2ff  ff,  (158) 

as  has  already  been  given  in  equation  (73),  page  42. 

From  the  triangle  A  B  C,  Fig.  1,  find  vl  in  terms  of  Fand  GO, 
giving 

v*  =    V*  _[_  r*  GO*  —  2  Fcos  «  .  ?\  GO,  (159) 

as  also  given  in  equation  (94)  ;  and  equation  (9)  gives 

(1  +  |i.)  «,'  =  <  +  (>-:  -  r?)  «•  ;  (160) 

equation  (11)  becomes 


and  (12)  and  (13)  give 


—  fi_r      r*  a,*  +   Fcos  a  -  r,  GO]  (162) 

9 

.-.E=  -J*      =    -Tr'C-n'  ^+  Fcos  «  '  r*  <»\ 
d  (J  H          g  1± 

which  is  a  maximum  for 

r,  GO  =  \  V  T±  cos  a,  (164) 

** 

in  which  1\  GO  will  be  the  velocity  of  the  outer  rim  =   V"  (say) 
and  if  the  buckets  are  very  narrow,  or  if  they  move  in  a   right 


148 


HYDRAULIC   MOTORS. 


line,  as  in  the  preceding  article,  or  practically  in  a  right  line,  we 
may  consider 

r,  =  r, ; 
then 

F"  =  iFcosar,  (165) 

or  the  velocity  of  the  bucket  will  be  one  half  the  velocity  of  the 
component  of  the  velocity  of  the  water  as  it  issues  from  the 
supply  chamber  in  the  direction  of  motion  of  the  wheel.  If 
a  —  0,  the  velocity  of  the  bucket  should  be  one  half  that  of  the 
jet ;  and  for  this  case  we  have 


v>  =  4  F.  (167) 

(1  +  /<„)  <  =  i  F2.  (168) 

F22  -  4  F;;:,,  (169) 

?7  =  LJ?  .j  V  =  %8  Q  H.        (170) 

^  =  i-  (171) 

Equations  (162)  and  (163)  show  that  the  work  and  efficiency 
are  independent  of  the  frictional  resistances  of  the  water,  but 
(166)  and  (168)  show  that  F  and  v.t  are  diminished  by  such  re- 
sistances. 

If  yz  be  finite  and  the  wheel  one  of  "  free  deviation,"  we  have 
from  equations  (158),  (159),  (11),  (12),  and  (13),  after  making 
dlf+  dtv&ty 

0  =  VI  +  ^  ( Vr,  cos  a  -  2  r?  GO)  + 

2  F2  —  3  Vr,  cos  of  .  GO  4-4  r  2  GO* 
r*  C°S  y*  V^T  -  2  Vr  cos  a    co  4-7^7  ' 

This  produces  a  complete  equation  of  the  fourth  degree,  and 
the  complete  solution  will  be  more  lengthy  than  for  the  pressure 
wheel,  as  given  in  equations  (16)  and  (60). 


FIG.  60. 


FIG.  61. 


HYDRAULIC     MOTORS.  149 

90.  -Cascade  Wheel.— The  James  Leffel  &  Co.  have  designed 
two  wheels  to  be  established  at  Ward,  Col.,  to  work  under  a  head 
of  730  feet.  One  is  to  be  38  inches  diameter,  to  be  driven  by 
one  nozzle,  producing  about  25  horse-power  at  about  552  revolu- 
tions per  minute ;  the  other,  50  inches  diameter,  fed  by  a 
nozzle  a  little  larger  than  1-J-  inches,  will  have  a  capacity  of 
about  140  horse-power.  Some  wheels  of  this  design  are  now 
working  under  high  heads.  Fig.  62  shows  one  of  these  wheels 
with  pulley.  In  actual  running  the  wheel  is  inclosed  in  a  case, 
which  was  removed  in  order  to  show  the  construction  of  the 
wheel. 


OF  THB 

UNIVERSITY 


INDEX. 


PAGE 

Angular  velocity 10, 19,  29 

Barker's  mill 48,  64 

Boott  turbine 89 

Boyden  diffuser 89 

Bryden  turbine 92 

Bucket,  depthof 89 

"      formof 17,86,123 

"      initial  angle  of 10,  14, 122,  133 

"      terminal  angle  of 122 

Buckets,  number  of 25,  122 

Centrifugal  force 5,  41 

Class-room  exercise 117 

Coefficient  of  effect 21 

Collins  turbine 94 

Comparison  of   inward  and  outward   flow 

wheels 35 

"    various  makes  of  turbines ..  130 

Crown,  width  of 121 

Crowns 27 

"       depth  between   72 

"       formof  125 

parallel 66 

Designing 3,27,117 

Diameter  of  wheel 120 

Diffuser,  Bryden 89 

Direction  of  quitting  water 9,  18,  84 

Efficiency 26 

"       effect  of  v  on 15 

"       equation  for 7,  28,  29 

"       maximum 8 

"         for  inflow 35 

"  u  "  outflow 35 

"       wheel  of  free  deviation 45 

Energy  imparted  to  wheel 13 

"      lost  by  impact 26 

"       "    in  escaping  water 26 

"      loss  of  due  to  quitting  velocity 35 

Escaping  water,  volume  of 73 

Exercises 25,  31,  41,  43,  46,  49,  70,  90,  117 

Fall,  power  of 117 


PAGE 

Form  of  bucket 17,86,123 

"      "  crowns 125 

Fourneyron  turbine 37,  93 

Francis  and  Thomson's  vortex  wheel .    37 

Free  deviation 18,  ^3,  44,  147 

1 '    surface,  form  of .  45 

Friction  along  buckets 26 

"      head  lost  by 4 

Frictional  resistance 5, 13 

Frictionless  wheel,  work  done  in 39 

"      efficiency  of 39 

"      path  of  water  for 40 

General  solution  of  pressure  turbine. . . 3 

Guides,  terminal  angle  of 10,  11,  35,  122 

Haenel  turbine 97 

Head  due  to  pressure  and  velocity 4 

"     lost  by  friction 4 

"     total 4 

"     virtual 3 

Hercules  turbine. „ , 107 

Humphrey  wheel 102 

Hurdy-gurdy  wheel 114 

Inflow  wheel,  efficiency  of 15,  26 

"  "      maximum  efficiency  of 35 

Initial  angle  of  bucket 10, 14, 122, 133 

"       velocity 4 

Inward  flow 9 

"  ratio  of  radii  for 25 

Jet  propeller 50 

Jonval  turbine . .  . .     37 

Leakage 137 

Maximum  efficiency 7 

Moment  of  stress 85,  141 

Momentum,  moment  of 56 

Niagara  wheel  . .  130 

Notation 1 

Number  of  buckets 25 

Outflow  wheel  efficiency  compared  with  in- 
flow     35 

Outward  flow 9 


INDEX. 


PAGE 

Outward  flow  efficiency  of . . . » 26 

"  "    ratio  of  radii  for 25 

Parallel  crowns 66 

flow  turbine. ./. 9,93 

Path  of  water 30,  85,  128 

Pelton  wheel 111 

Poncelot  wheel  J14 

Pressure  at  entrance  to  bucket 23 

"        "  exit  from  bucket 6 

"       due  to  deflecting  stream 143 

"       in  wheel 22,  82 

"       theoretical  at  entrance  to  bucket. ..      4 

"       turbines,  general  solution  of 3 

Quitting  angle,  value  of 9,  18,  84 

Radii,  ratio  of 25 

Rankine  wheel 37,  65 

Relation  between  y2  and  w1 10 

Revolutions  for  best  efficiency 47 

Risdon  wheel 101 

Scottish  and  Whitelaw  turbine 50 

Segmeutal  feed 95,  96 

Shaft,  diameter  of 85 

Stress,  moment  of 85,  141 

Submerged  wheels .23,  24 

Swain  turbines 103 

Tables,  effect  of  different  values  of  y1 14 

"        "  Y!  on  velocities 15 

'         for  revolutions  and  H.P.  for  best 

efficiency 47 

Tangential  wheels . .     99 

Terminal  angle,  effect  of  large 78 

"  "      of  bucket 122 

"      "guide 10,11 

"       velocity 4 

Tests,  Boott  turbine   90 

"      Bryden  turbine 92 

"      Collins  turbine 94 

"      Haenel  turbine 97 

"      Swain  turbine 105 

"      Tremont  turbine 68,  69 

'-'      Victor  turbine  ... UO 

Total  head 4 

Tremont  turbine  — 67 

Turbine,  Boott 89 

Boyden 92 


PAGK 

Turbine,  Collins 94 

"        Fourneyron.  . . .   37,  93 

cut 4 

triple,  cut 10 

"       Guideless,  cut 24 

"       Haenel 97 

"       Hercules 107 

' '       Inflow,  cut 12 

"       Jet,  cut 26 

"        Jonval 37,  93 

"        mixed  flow,  cut 20 

"        outflow,  cut ....      2 

"       parallel  flow  93 

"  "          "  cut 14.  16 

"       Scottish  and  Whitelaw 50 

"        Swain 103 

"       Tangential,  cut 22 

"       Tremont G7 

Victor 109 

Values  of  a , 11 

"«  +  7i 10 

"  e is 

"  MI  and  M» 13,84,  70,  71,78,  138 

"  <o' 19 

Velocity  along  rotating  tube 41 

"       at  exit 6 

"        of  entering  water .  .8,  28 

"         "  initial  rim 12 

"         "  quitting  water 9,  28 

"         "  terminal  rini 12 

"         "  wheel  for  best  effect 10 

Velocities,  initial 4 

"  terminal 4 

Victor  turbine 109 

Virtual  head 3 

Volume  escaping  water 73 

"      of  water  flowing  through  wheel  —  136 

Vortex  wheel,  Francis  and  Thomson's 37 

Wheel  of  free  deviation 44,  147 

Work,  equation  for 7 

' '       done  by  centrifugal  force 5 

"  "    "   falling  weight 5 

"    upon  wheel 6 

*'       lost  by  friction 32 


SHORT-TITLE   CATALOGUE 

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